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CORNELL

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MATHEMATICS LIBRARY

Cornell University Library
QA 47.B86R4

Report

3 1924 001 918 857

Cornell University
Library

The original of tiiis book is in
tine Cornell University Library.

There are no known copyright restrictions in
the United States on the use of the text.

http://www.archive.org/details/cu31924001918857

REPORTS

ON

THE STATE OE SCIENCE.

Report of the Committee, consisting of Professor Cayley^ F.R.S.,
Professor Stokes, F.R.S., Professor Sir W. ThomsoisTj F.R,S.,
Professor H. J. S. Smith, F.R.S., and J. W. L. Glaisheu, B.A.^
F.R.A.S. (Reporter), on Mathematical Tables.

§ 1. General Statement of the Objects of the Commiitee,

The purposes for which the Committee was appointed were twofold, viz,
(1) to form as complete a catalogue as possible of existing mathematical
tables, and (2) to reprint or calculate tables which were necessary for the
progress of the ma,th€matjcal sciences. • . ■ , » ■,

These two objects, although so far connected, that it was absolutely
essential before any tables were calculated or reprinted to be certain that
such tables were not already in existence or easily accessible, were in other
respects quite different ; and the Committee have therefore decided to keep
them distinct. The reasons in favour of the adoption of this course are ob-.
viously very strong, as a new table would be out of place in a Keport which
in other respects was merely a detailed catalogue. A further argument
against the publication of the tables in the Eeports of the Association, is
the great objection to needlessly scattering tables. Tables of a kindred
nature collected together, are of far more value than the same could be if
dispersed in several volumes of a periodical ; and if the tables of the Com-
mittee were published annually as calculated, it would happen not only that
they would have to be sought in several volumes, and their utility in conse-
quence considerably impaired, but sometimes even portions of the same table
would be separated, The Committee have therefore considered that they
would best carry out the second object for which they were appointed, by
publishing their tables separately and independently of the Annual Eeports
of the Association.

The form chosen for this publication is a quarto of the same sizfe as that
of &e Philosophical Transactions, this size being necessary for the uniformity
t)f the tables, as g, large page is required in order to contain the values of th?
function tabulated, together with its first, second, and third differences, which,
when given, should range with the fonner on the same page. Before the

1873. - >' B

2 KEPORT— 1873.

appointment of the Committee, certain tables of hyperbolic antilogarithms or
exponentials (viz. e^ and «~*) and of hyperbolic sines and cosines had been
commenced by Mr. J. W. L. Glaisher ; and these the Committee determined
to print and stereotype on their completion. They are now in the press.
A mass of calculations has been made for the tabulation of Bessel's functions,
for real and imagiiiary values ; and it is intended to complete these tables, and
then to undertake calculations connected with the Elliptic Functions.

As yet no tables have been reprinted by the Committee ; and it clearly
would not be possible to decide which most required reproduction, until the
Eeport was considerably advanced beyond its present stage.

All the tables printed by the Committee, whether calculated or reprinted,
are to be stereotyped ; and it is intended that they shall ultimately form a
volume ; but the tables relating to each function will be pubUshed and circu-
lated separately as calculated, the stereotype-plates remaining in the posses-
sion of the Committee for future use.

The first object of the Committee was rendered necessary by the fact that
the mathematical tables that have been formed, are scattered all over the
world in the various mathematical and scientific journals, transactions of
societies, &c., so that it is extremely difficult to ascertain what tables have
been already calculated in any particular branch of science. Another reason is
that tables formed for some particular purpose, and published under a title of
special application, are often of equal importance in other investigations ; so
that great inconvenience is sometimes felt for the want of a table which
already exists under another name and having reference to a different subject ;
or it may even be recalculated. The difficulty of knowing exactly the work
already done in any subject is one which is common to all parts of science ;
but the inconvenience resulting from the nature of a work being obscured by
its name is to a great extent peculiar to this subject, or at aU events is more
painfully felt in connexion with it. A familiar instance of a function occurring

/loo

in several distinct subjects is the integral Ig-^^c^a;, which is of imfortance

in the determination of the probable error in the method of Least Squares,
Astronomical Eefractions, and the theory of Heat; and good instances of
the manner in which the nature of a table can be obscured by its name are
affijrded by nautical collections, where under such headings as " Table to
find the latitude by double altitudes of the sun and the elapsed time," or
" Table of logarithmic risings," &c., are given log cosecants, log versed sines,
&c. A Catalogue, therefore, in wldch the tables were carefully described
from iheir contents seemed very desirable ; and this the Committee hope to be
able to accomplish by their Reports.

It is intended to include all numerical tables that can be regarded as
belonging to mathematical science, or which are of interest in connexion
therewith ; but none wUl be noticed in which the tabular results or data are
derived from observation or experiment, or merely concern special subjects
that are not generally classed under the head of mathematics. Thus the
great majority of astronomical tables, including catalogues of stars, tables of
refraction, tables depending on the figure of the earth, &c., will 'be ex-
cluded, as the data for the formation of such tables are derived from observa-
tion. The same remark applies to all chemical tables, tables of specific gravity,
of weights and measures, for the determination of the longitude at sea, mortality
tables, &c. Life-assurance and annuity tables, and all commercial tables
wiU also be excluded. With regard to these last, however, although all tables
such as ready reckoners and common interest tables will in general be omitted.

ON MATHEMATICAL TABLES. 3

any one that is of value in. relation to mathematics as a science wiU be in-
cluded, although it may have been calculated for merely commercial purposes
and published under a name that would apparently eiclilde it from this Eeport.
Many tables of compound interest are valuable when viewed as tables of powers j
and many navigation tables calculated merely for the use of the sailor, and pub-
lished under titles that would imply that they were of a merely technical cha-
racter, are in reality trigonometrical tables Under a disguised form.

From the above rematks it will be found in most cases very easy to decide
whether a table is included in the scope of this Eeport or not. A few of ooiitse
come on the boundary ; and then there is some little difGlculty in drawing the
line fairly. Of this kind are tables for the expression of hours and minutes as
decimals of a day, &c. ; most of these it has been thought better to include.

It was necessary as a preliminary to form a classification of mathematical
(numerical) tables ; and the following classification was drawn up by Prof*'
Cayley and adopted by the Committee.

A . Auxiliary for non-logarithmic computations.

1. Multiplication.

2. Quarter-squares.

3. Squares, cubes, and higher powers, and reciprocals.

B. Logarithmic and circular.

4. Logarithms (Briggiah) and antilogarithms (do.) ; addition and sub-

traction logarithms, &c.

5. Circular functions (sines, cosines, (fee), natural, and lengths of circulai*

arcs.

6. Circular functions (sines, cosines, &c.), logarithmic.

C. Exponential.

7. Hyperbolic logarltlims.

8. Do. antilogarithms (e*) and li . Itail (45°+^^), and hyperbolic sines,

cosines, &o., natural and logarithmic.

D. Algebraic constants.

9. Accurate integet dt fractional values. Bernoulli's NoS., A" 0™, &c,

Binomial coefiicients. , ^ ^

10. Decimal values auxiliary t6 the calculation of sefies.

E. 11. Transcendental constants, e, tt, y, &t., and their powers and functions,

F. Arithmological.

12. Divisors and prime numbers. Prime roots. The Canon anthmeticus&d.

13. The Pellian equation.

14. Partitions.

15. Quadratic forms a^+h^, &c., and partition of numbers mtd squares,

cubes, and biquadrates.

16. Binary, ternary, &c. quadratic and higher forms.

17. Complex theories.
Gr. Transcendental functions.

18. Elliptic.

19. Gamma. . i ; , i

20. Sine-integral, cosine-integral, and exponential-integral.

21. Besscl's and allied fimotions.

22. Planetary coefficients for given -,.

23. Logarithmic transcendental.

24. Miscellaneous.

B 2

4 REPORT — 1873.

Several of these classes need some little explanation. Thus D 9 and 10 are
intended to include the same class of constants, the only difference hcing that
in 9 accurate values are given, while in 10 they are only approximate ; thus,
for example, the accurate Bernoulli's numhers as vulgar fractions, and the
decimal values of the same to (say) ten places are placed in different classes, as
the former are of theoretical interest, while the latter are only of use in calcu-
lation. It is not necessary to enter into further detail with respect to the
classification, as in point of fact it is only very partially followed in the Report ;
the final index, however, will be constructed as much in accordance with it as
possible.

The only perfect method by which all the tables on the above subjects could
be found with any certainty, is to examine all the volumes of the mathema-
tical and philosophical journals and transactions, given in the list prefixed to
the Eoyal Society's Catalogue of Scientific papers — a most laborious work, as
it requires every page in all these periodicals to be looked at, and any nu-
merical tables noted and subsequently examined, while if included in the
scope of the Committee's work they must further be described. The mere
turning over the pages of several thousand volumes is a work of some labour,
and the completion of the Report must occupy the Committee for several
years. The work is also of such a nature that it would not be possible to
obtain even an approach to completeness in any one class till very considerable
progress had been made with the preliminary examination.

This, however, is not the case to any great extent with the groups A and
B, or with C 7 or the first part of F 12, as tables in these classes are gene-
rally to be found in separate books, and not in the memoirs of societies, or
journals. It was possible, therefore, to make progress in the above classes
immediately ; and the portion of the Report now presented to the Association,
practically contains a catalogue of tables which form separate books. The
three broad divisions into which mathematical tables divide themselves
practically are found to be : —

I. Subsidiary tables, which are rather of value as a means of performing
calculations than of interest in themselves : e. g. multiplication tables,
logarithms, &c. They generally form separate books.

II. Tables of continuous functions, generally definite integrals.

III. Tables in the theory of numbers.

Divisions II. and III. contain conclusive (in opposition to subsidiary)
tables.

A fuller description of the contents &c. of Division I. will be found in
§ 2. It is hoped next year to report on Division II., and the next year on
Division III. It will be necessary afterwards to add supplements to different
classes, and notably to the present portion of the Report, which has no claim
at all to be regarded as complete, but is published on the distinct understand-
ing that it is by no means exhaustive with regard to the subjects treated in
it ; a supplementary Report on the same subject wiU be subsequently added ;
and it is hoped that thus it will be rendered complete (see § 2).

§ 2. General Introduction to the present Report, and Explanation of its
Arrangement and Use.

Art. 1. The present Report is intended to include all general tables, viz.
tables that are of general application in all branches of matJiematics, and
are therefore useful wherever calculations have to be performed. The most
simple instances are multiplication tables, common logarithms of numbers,

ON MATHEMATICAL TABLES. ■ 5

and trigonometrical functions, Avliich. form the basis of, and are the means
by which all other calculations are made, llegarded from this point of view,
this division may be said to contain auxiliary or subsidiary tables, viz. such
as arc not per se of any very great intrinsic interest (multiplication tables
are a good instance), but which are nevertheless of such paramount import-
ance that, without their aid, the calculation of other tables would be too
laborious to be practicable. As before remarked, one reason why these tables
may well form a division by themselves is, that, being intended for calcula-
tions of all kinds, they are usually published separately, and have not to be
sought among the transactions of societies and other periodicals. The num-
ber of tables in this class is of course many times greater than are aU the
other classes put together ; but then, on the other hand, they admit of more
brief description, ■ as scarcely any explanation is needed of the functions
tabulated, or of the purposes for which the calculation or publication was
undertaken. In the present Keport not above five or six tables printed in
periodical publications are noticed ; while it is probable that in the Eeports
on the other classes there will not be a much greater number that will have
appeared as separate and independent books.

Art. 2. The object of the Eeport is to enable any one by means of it to
find out with ease what tables have been computed on any of the twenty-
five subjects (see § 3) to which it relates, and where they are to be found ;
and the desire to form a catalogue that shall give a systematic and practical
account of the numerical tables in existence that bear upon each of the
subjects included has been steadily kept in view ; in fact little else has been
aimed at. Still, as in the search for and examination of so many books of
tables (the Report contains an account of more than 230) a good many works
of considerable historical or bibliographical interest came to light, it was not
thought desirable to suppress all notice of them. The majority of seven-
teenth-century works included are described, on account either of their rarity
or because they serve to illustrate the history and progress of the subject.
Of this kind are I^apier's ' Canon Mirificus ' (1614), containing the first an-
nouncement of logarithms, Ltoolf's 'Tetragonometria' (1690), &c. ; and when
such works have been included, their full titles have been given in § 5, with
suitable bibliographical accuracy. It would be a mistake, however, to suppose
that all the tables of the seventeenth century have been superseded ; Viacq's
' Arithmetica,' 1628, is the most convenient ten-figure table of logarithms
that exists (it has only been reprinted once, and not in so useful a form) ; ' and
no natural canon published subsequently can bear comparison with Pitisous,
1613. In performing mathematical calculations, we have had repeated occa-
sion to use both Vlaoq and Piiiscirs. Uesintjs's ' Napierian Canon' (1624)
is the largest in existence. The points in which the Eeport is least complete
are the descriptions of common tables of the eighteenth century, and of com-
paratively modern Italian, Spanish, &c. tables of logarithms. The former
class we have purposely omitted, though we have examined many, as they
are neither of value intrinsically nor historically; a good many are briefly
noticed by De Morgan ; and the latter we have not been able to see : several
titles will be found in the Babbage Catalogue.

Art. 3. The most valuable detailed list of tables hitherto published is the
article Tables written by De Morgan for Knight's ' English Cyclopsedia '
(1861). This article first appeared in the ' Penny Cyclopsedia' (1842), but
it was carefully revised and largely augmented by its author before its re-
printing in the ' English CyclopEedia.' In this article are contained notices
of 457 tables, many of which, however, are outside the scope of this Eeport.

6 . REPORT— 1873.

We have had occasion to make great use of this article ; and whenever De
Morgan's name is cited without reference to any work of his, it is always to
he understood that it is this article which is referred to. Other works which
we have used, but which contain information almost whoUy of a bibliogra-
phical or historical nature, are : —

(1) ' Historia Matheseos Universse a mundo condito ad seculum P. C. N.
SLVI. . . . accedit . . . historia Arithmetices ad nostra tempera,' autore Jo.
Christoph. Heilbronner. Lipsise, . . . 1.742, 1 vol. 4to. The « Liber quartus
sistens Historiam Arithmetices' is at the end of the book, and occupies
pp. 723-924.

(2) ' Geschichte der Mathematik,' von Abraham Gotthelf Kastner, Got-
tingen. (4 vols. 8vo, 1796-1800.) It forms the seventh 'Abtheilung' of
the ' Geschichte der Kiinste und Wissensohaften ' (57 vols,). The tables are
contained in vol. iii.

(3) ' Bibliotheca Mathematica,' auctore Frid, Guil. Aug. Murhard. Lipsise,
1797-^1804 (also German title, ' Litteratur der mathematisohen "Wissen-
schaften'). 4 vols. 8vo. ' Mathematische Tafeln' is the heading of the
fourth division of vol. ii., and occupies pp. 181-201 ; they are divided into
two classes, the first containing logarithmic and trigonometrical tables, and
the second the rest; works that Murhard has hsvd ju his own hands are
marked with an asterisk.

(4) ' Bibliotheca Mathematica sive Criticus Librorum Mathematioorum,
. , . , commode diapositus ab J. Eoggio.' .Sectio I. ' Libros Arithmetices ct
Geometricos complectens.' Tubingee, .... 1830 (also with German title-
page). This work we have found very useful, A great number of logarithmic
and trigonometrical tables are carefully described in Div. IV. ' Elementar-
Geometrie' (B.), pp. 367-410. It is right to add that the titles of tables
are to be found in all portions of the work, and are by no means restricted
to the arithmetical divisions. We believe that no more than the ' Sectio I,'
was ever published.

The following is a continuation of Bogg : — ■

(5) ' Bibliotheca Mathematica, Catalogue of Books in every branch of
Mathematics .... which have been published in Germany and other coun-
tries from the year 1830 to the middle of 1854.' Edited by L. A. Sohnke,

. . . with a qomplete index of contents. Leipzig a,nd London, 1854, 1 vol.
8vo.

(6) ' Bibliographic Astronomique, avec I'histoire de I'Astronomie. . . . Par
Jerome De La Lande ... A Paris. ... An XI. = 1803, 1 vol. 4to. A sepa-
rate index to the general tables is given on pp. 960, 961.

(7) ' Litteratur der Mathematik, Natur- und Gewerbs-Kunde mit Inbegriff
der Kriegskunst,' . . . von J. S. Ersch. ' Neue fortgesetzte Ausgabe,' von F,
W. Schweigger-Seidel. ' Aus der neuen Ausgabe des Handbuchs der Deuts-
cheu Litteratur besonders abgedruckt.' Leipzig, 1828. 1 vol. 8vo.

(8) ' Biographisch-literarisches Handworterbuch zur Geschichte der exactcn
■VVissenschaften . . . gesammelt' von J. G. Poggendoff. Leipzig, 1863. 2
vols. 8yo.

(9) ' E. P. Claudii Francisci Milliet Dechales Oamberiensis e Societate
Jesu Cursus seu Mundus Mathematicus.' . . . Lugduni, 1690. 4 vols. fol.
The first volume opens with a ' tractatus Proemialis de progressu Mathe-
seos et Ulustribus Mathematicis ; ' and pp. 28-37 are devoted to arithmetical
bibliography. We may state that a previous edition of 1674, in 3 vols, fol.,
does not contain the ' De progressu.'

ON MATHEMATICAL TABLES.

We may also mentioa De Morgan's ' Arithmetical Books from the invention
of printing to the present day,' London, 1847, 8vo, the introduction of which
contains useful bibliographical information about the description of books,
and Peacock's " History of Arithmetic " in the ' EncyolopEsdia Metropolitana.'
. There is one bibliographical work, viz. Soheibel's ' Einleitung zur mathe^
matischen Bueherkenntniss.' Neue Auflage. 3 vols. 8vo, Breslau, 1781
(as given in the Babbage Catalogue), which ia continually referred to by
Murhard, Bogg, &c., though we have never been able to see a copy in any
library to which we have had access, or procure one otherwise. De Morgan
says, "Soheibel (additions) may be considered as partly repetition, partly
extension, of HeUbronner. He is one of those bibliographers who collect
from, various sources the names and dates of more editions than those who
know catalogues will readily believe in."

It is unnecessary here to mention works on general bibliography, such as
Hain, Ebert, Watt, &c., which are well known ; we may, however, partis
cularly notice 'Tresor de livres rares et precieux ou Nouveau dictionnaire
bibliographique,' par Jean George Theodore Grraesse, Dresde [also Geneva,
London, and Paris], 1859-1867 (7 vols, including supplement), which might
be of use, though we have found tbe mathematical works it contains very
inaccurately described ; but this is a fault oomnion to all works of general
bibliography.

Montuda, ' Histoire des Mathematiques,' we have not found valuable ; but
we may call attention to the accurate information given by Delambre in his
' Histoire de I'Astronomie Moderne,' t. i, Paris, 1821 ; and alsq in his other
histories.

Keuss's ' Eepertorium Commentationum a sooietatibus litterariis editarum,'
Gottingae, 1801-1821, 16 vols. 4toj is a work veiy similar in its plan to
the Hoyal Society's Catalogue of Scientific Papers, except that it is an indetn
rerum inateai of an index auctonim. The mathematics is contained in vol.
vii., the arithmetic occupying pp. 2-31 of that volume. On p. 30 are refer-
ences to descriptions of calculating and other arithmetical machines.

We have found Nos. XIX. and XX. (on trigonometrical and logarithmic
tables) of Button's ' Mathematical Tracts,' London, 3 yols. 8vo, 1812, very
useful.

Art. 4. The mode of arrangement of this Eeport (which properly occu'
pies § 3, § 4, and § 6), and the reasons that have led to its adoption, are as
foUows : — If every table were published separately and formed a work by
itself, the obvious course would be to divide them into a certain number of
classes according to their contents, to prefix to each class a brief intro-
duction and explanation, and then to give a detailed description, in chrono-
logical order, of the tables included under it. This is, in fact, the course
that has been pursued with regard to separate tables (i. e. works containing
either a single table or only tables that come under the same class) ; § 3 is
divided into 25 articles, each article being devoted to one subject ; — art. 1,
multiplication tables ; art. 2, tables of proportional parts, &o. (for the con-
tents of all the articles, see the commencement of § 3). Each article begins
with a general] account, partly historical, of the subject iucluded in it ; and
then follow the" descriptions of the separate tables on that subject. But the
majority of works noticed are collections, and include tables that are com-
prised under several articles; thus Hutton's tables contain Briggian and
hyperbolic logarithms of numbers, a natural and logarithmic canon, &c. &o.,
each of which belongs to a diiferent article. Two courses were therefore
open for the treatment of such works : — (1) to describe them under the article

8 REPORT— 1873.

having reference to the first or largest table in the work, and insert cross
references under each of the articles concerned with the other tables in-
cluded in the work ; or (2) to describe all collections of tables in a section
by themselves, and give references to each of the tables they contain under
the appropriate article in § 3. The second course was clearly the more
proper, for three reasons' — (1) because it was free from the arbitrary element
involved in the choice of the leading table, which would be required in tho
first method, (2) because it was undesirable to overload the articles of § 3
with descriptions of tables not belonging to them, and (3) because reference
to the works would be greatly facilitated by placing them in an article by
themselves ; § 4 therefore contains all works the contents of which do not
belong wholly to one of the articles in § 3, or, in other words, wliieh con-
tain at least two tables, the subjects of which are included in different
articles of § 3. As the works in § 4 will thus have to be continually re-
ferred to separately, they are arranged alphabetically, not chronologically.
§ 5 is a complete list of all the works containing tables that arc described
in this Report ; and to facilitate its use as an index, a reference is attached
to the section, or section and article, in which the work is described.

To take an example of the manner in which the Eeport is intended to be
used. Supposing it were required to know what tables there were of log
versed sines ; the reader would turn to the beginning of § 3, and, looking
down the list of articles, see that, coming under the head of " logarithmic
trigonometrical functions," such tables belonged to art. 15. He would ac-
cordingly turn to art. 15, and read or glance through the introductory
remarks to that article, and the works described there ; not finding any book
containing log versed sines alone described in the article, he would conclude
that no separate table of the kind had come under the notice of the reporter ;
he would then look at the references to § 4 ; and if he wished for detailed
information with regard to any of those tables, he would examine the de-
scriptions in that section. Any one, on the other hand, desiring to know
the contents of any particular work would seek it in § 5 ; if it occurred there,
a reference would be foimd added either to § 4, or to § 3 and the article in
which it is described. No difficulty wiU be experienced in. finding the descrip-
tion if it be remembered that all the works are cited by the author's name and
the date ; and that while in § 4 they are arranged alphabetically, in the articles
of § 3 the arrangement is chronological.

The date is throughout appended to the author's name in citing a work, in
order to identify the work in § 5 (the date given being always that assigned
to the work in § 5) ; there is also the further advantage, that any one who
requires information only with regard to modern tables, still procurable from
the bookseller, need not waste time in seeking the detailed descriptions of
Avorks published in the seventeenth and eighteenth centuries.

It may be mentioned that a few works that do contain tables of more than
one kind, are nevertheless included in § 3 : this happens when the smaller
tables are insignificant compared with those under which the work is classed ;
references are then appended also in the articles to which the smaller tables
belong.

An asterisk prefixed to an author's name (thus * Voisin or * Voisin) in-
dicates that the description of the work of his referred to has not been derived
from inspection. In every case where there is no asterisk, the description
has been written by the reporter with the book itself before him.

Art. 5. In aU cases where the author of a collection of tables has num-
bered or marked them himself, his numbering or marking has been followed

ON MATHEMATICAL TABLES. y

in this Report, except in very exceptional circumstances. Where, however,
the tables are not numbered or otherwise denoted, they have been marked
[T. I.], [T. II.], &c., as it was necessary to have the means of referring to
them. Invariably, therefore, where the number of the table is not included
in square brackets, it is to be understood that it is the author's own number.
Thus T. VII. in any particular work implies that the table in question is
numbered YII. in that work, while [T. YII.] implies either that the table
has no number, or that the classification in the work is different from that
adopted in this Eeport. Whenever logarithms are mentioned without the
epithet hyperbolic or Napierian, common or Briggian logarithms (viz. to base
10) are intended. In some cases, where there might be some doubt, the
adjective "common "is introduced. By hyperbolic logarithms are always
meant logarithms to the base e (2-71828 . . . ); and these are never called
Nwpurian, this word being reserved for logarithms of exactly the same kind
as those introduced by Napier (see § 3, art. 17). Such a sentence as " Five-
figure logarithms to 1000," ia always to be understood as meaning " logarithms
of numbers from unity to 1000, at intervals of unity to five decimal places :"
viz., when the lower limit of a table is not expressed, it is always to be taken
as unity ; and when the intervals are not mentioned, they are always unity.
The term " places " is used throughout for " decimal places " or " decimals,"
a number " to 3 places " meaning a number given to 3 places of decimals
(not Z figures'). The only exception made to this rule is in the description of
tables of common logarithms ; the words " seven-figure logarithms, gix-flgure
logarithms," &c., have become by usage so completely recognized as meaning
logarithms to seven places, to six places, &c., that it did not seem worth while
disturbing the established mode of expression, as it could lead to no error.

The contents of old works have been described in the language and nota-
tion of the present day, and not in the manner adopted by their authors ;
any peculiarities of notation &c. in a table, however, are pointed out. It was
long universal, and is still very common, to describe trigonometrical tables as
being computed to a certain radius; these are translated into the language
of decimals ; thus a table " to radius 10,000,000 " is described as a table
" to seven places," and so on. As a rule the characteristics of the logarithms
have been ignored in describing a table ; i.e. it has not been stated whether
the characteristic was given or no, or, if given, what was the understanding on
which it was added. In many tables, contained in works intended for a special
purpose (as in collections of nautical tables, &c.), arbitrary numbers are added
to or subtracted from the characteristics to facilitate their use in working
some particular formula ; to have included details of this kind would have
taken much room, and been really superfluous, as in most cases all that is
required to be known in the description of a table of logarithms, is the range
of the table, and the number of places to which the mantissse are given.

We may here mention that an ambiguity occurs in the description of propor-
tional-part tables ; thus a " table of ijroportional parfs to tenths " may mean
either that the proportional parts are given for one, two, three, &c. tenths of
the difference, or else that the numbers that form the proportional-part table
are given to one place of decimals. The former is the meaning generally in-
tended ; and it would be better if in this case the words " to tenths " were
replaced by " for every tenth.'" . , , ^ ^^ ^. .^ ,. :,

A good many tables had been described before the ambiguity was noticed ;
but it is believed the context will generally show the true meaning ; when
the words to tentJis, to hundredths, &C. are italicized, the latter interpreta-
tion (viz. results given to one, two, &c. decimal places) is to be assigned.

10 REPORT— ISrS.

Art. 6. To the particular editions of the works described no importance
is to be attributed. It would obviously have been impossible to always fix
upon the first or last edition as the one to be described ; in fact we had no
choice ; we took what we could get. The list in § 5 always contains portions
of the titlepage of the same edition of the work that is described in § 3 or
§ 4 of the Eeport ; the particular edition chosen was usually determined by the
accidental circumstance of its being the first that was examined, any informa-
tion that was subsequently obtained about other editions being added at the
end of the description of the contents of the work in § 3 or § 4. It would
have been better to have always taken as the standard the last edition pro-
curable, and pointed out wherein it diflfered from its predecessors j but this
would have required much rewriting of particular portions, and considerably
increased the labour of preparation, with a very small increase of regularity
in the aiTangement of the Report, but with no corresponding increase in its
value.

Art. 7. In every case where a table has been described from inspection, all
the tables themselves have been examined, and not merely their- titlepages,
tables of contents, &c. This was of course absolutely necessary in very many
instances, as it is comparatively rare that any thing more than a general
notion of the contents of a collection of tables can be gathered from the author's
explanations ; but in any case it was essential if the Eeport was to have any
value for accuracy, because the titles assigned by their authors were sometimes
misleading, if not absolutely erroneous ; and frequently, even if the more im-
portant tables had headings or descriptions prefixed, the smaller ones (which are
often more worthy of notice on account of their rarity or mathematical value)
were passed over. It must here be remarked that it is never safe to take
a description of a table from its author or editor, as it is not a very uncommon
thing to give as the contents of a table, not that which can be found from it at
once, but what can be obtained from the table by means of additional work,
such as an interpolation. Thus, under the heading " Table of logarithms to
eight decimals " is sometimes given a table to five places, and a formula from
which to calculate the remaining three.

Another case in point is Steinbergeb's table, described in this Report, the
titlepage of which describes it as giving the logarithms of all numbers to
1,000,000, when in point of fact it only extends to 10,000 — tho justification
for the title being that two more figures can be interpolated for. It is not
to be supposed, when such misstatements occur, that the author of the table
has any desire to mislead, as they usually result from ignorance ; but it is a
matter of regret, when it has become customary (and most properly so) that
a table should be described on its title as giving only what can be taken out
of it without additional calculation, that this rule should sometimes be vio-
lated and a designation given that is, to say the least, misleading. We have
also met with such instances as the following : — The title of a book is given
id a bookseller's catalogue as (say) " Table of divisors of numbers from 1 to
10,000,000 ;" but (the following words (say), " Part I. from 1 to 150,000 "
(when perhaps no more was ever published), are left out — an omission of
rather an important character as regards the contents and value of the table.
Gases of this kind show how imperatively necessary it is to examine the
table itself; and whenever the description of a table is taken from an adver-
tisement, bookseller's catalogue, or other second-hand source, there is great
liability to error.

Art. 8. The names of authors occurring in tho text have been printed in
small capitals when the work of theirs alluded to is described in this Report,

ON MATHEMATICAL TABLES. 11

otherwise in ordinary roman type : thus we should write " the table was
copied from ' Briggs's ' Aritbmetica ' of 1624," because an account of Beiggs's
work is given in the Eeport ; but we should write " the sines were taken from
Yieta's ' Canon ' 1579," because Vieta's work is not described. This rule is
attended to always whenever an author's name is mentioned in juxtaposition
with his work, and it wUl he found to save unnecessary trouble in searching
for works not noticed in the Eeport. Of course aU rules are sometimes diffi-
cult to carry out ; and in cases such as when the author's name and work are
separated from one another, or the name occurs frequently in a paragraph by
itself, but really in connexion with some work not expressly named each time,
&o., we have attempted to carry out the spirit of the rule and no more. An
author's name is enclosed in square brackets (thus [Pell] or [Peli,]) when
his name does not occur on the titlepage of the work of his referred to.

Art. 9. The words 8vo, 4to, &c. are used in § 5 to signify works of
octavo, quarto, &c. size, without reference to the number of pages to the sheet.
They are merely intended to give a rough idea of the size and shape of the work,
which is better done by using them in a general sense than by attaching to
them their technical meanings. The words " large " or " smaU. " have been
prefixed when the size was markedly different from what is usual. It must
be remembered that two hundred years ago all the sizes were much smaller
than at present, so that the usual quarto page of 1650 is smaller than au
octavo page of our day, though the shape is of course more square. Old works
are generally described as they would have been at the time ; but it sometimes
may have happened that a true quarto of old date is here given as octavo, &c. :
this caution is necessary for those who might use § 5 bibliographioally.
Whenever, in transcribing portions of works in § 5, words have been omitted
from the titlepage, dots have been inserted to mark the omissions. We may
mention that we have used the word reprint in its proper sense ; viz. we have
not spoken of a reprint except when the type was reset.

Art. 10. In the preparation of this Eeport extensive use has been made of the
libraries of the British Museum, the Eoyal Society, the University of Cam-r
bridge, the Eoyal Observatory, Trinity College (Cambridge), and the Eoyal
Astronomical Society, in one or other of which the majority of the works
noticed are contained. We have also, through the kindness of Professor
Henrici, been enabled to consult the Graves Library at University College,
London, which contains an almost unrivalled collection of old mathematical
works ; but as they are not yet arranged, it is not possible to find any par-
ticular work without great expenditure of time and labour. The De-Morgan
library at the London University is also stiU in process of arrangement, and is
therefore inaccessible for the present. By the kindness of Mr. Tucker, who
forwarded us an early copy of the sale-catalogue of the late Mr. Babbage's
library, we have been enabled to extract several titles from it, and identify
works of the titles of which we had only imperfect descriptions ; but we have
not been able to see any of the books themselves. It must not be understood
that the Eeport contains notices of all the books of mathematical tables
contained in the libraries mentioned at the beginning of this article. For in-
stance, the Eoyal Society's catalogue contains the titles of several works that
should' be included but which we have not yet examined; and of course no
one can know what tables there are in such libraries as those of the British
Museum or the Cambridge University, where there is no catalogue of subjects.
-Por the omissions we could have rectified we must plead in excuse the
already great extent of the Eeport, and consequent necessity of drawing the
line somewhere. Of course many of the works noticed are either in oiu: own

12 REPORT -1873.

possession or were lent by friends ; and we must acknowledge the kind nssist-
anoe rendered by Mr. C. "W. Merrifield, P.R.S., of whose mathematical library
we hope to make more use in a future Keport.

Art. 11. The Eeport is avowedly very imperfect; it contains probably not
one half of the works that have as good a right to be noticed as those that
are included. This defect will be i-emedied by the publication of an Appen-
dix or additional Keport on the same subject, probably after the appearance
of the Reports on the other divisions. As it would he clearly impossible to
have made this Eeport perfect (and had it been possible, it would have occu-
pied more space than could be given to it), an Appendix giving the results
of the examinations of the memoirs, transactions, <&c. in reference to this
class of tables would have had in anj' case to be added after the com-
pletion of the other divisions ; and on this account it seemed unnecessary
to take especial pains to procure works that were clearly of no very great
importance, or to insert imperfect second-hand accounts of tables that would
in all probability be met with in the course of the formation of the subse-
quent Reports. Invariably, however, whenever a reference was found to a
table that seemed of importance, no pains have been spared in the endeavour
to obtain and examine a copy ; in the event of these efforts being fruitless,
a notice of the work compiled from other accounts has been given, with an
intimation of the source whence the information was derived ; but only three
or four works are included that have not come under the eye of the reporter.
It is probable that there may have been published recent works on the
continent no copy of which is contained in any of the public libraries of this
countrj' ; and on this account it will probably be found very difficult to
make the list perfect. The present Report is, however, so far complete that
the Committee think they may ask mathematicians or computers who arc ac-
quainted with any works not included in it or in De Morgan, to inform them
of the fact. It is only in this way that completeness can be obtained, as
although, by an examination of the transactions &c. to which references are
given at the beginning of the Royal Society's catalogue, the completion of the
accounts of tables contained in memoirs &c. would be merely a matter of time
and labour on the part of the members of the Committee, the discovery and de-
scription of books printed in out-of-the-way places, or for private circulation,
can only be effected by the cooperation of mathematicians who may happen
to possess copies*. The Report, however, as it now stands, will be found to
contain more information about tables than is to be found anywhere else ; in
fact, except De Morgan's list (referred to in art. 3 of this seetion), we know no
place where any attempt is made to cover the ground included in this Eeport ;
and though De Morgan has referred to more works than are described here in
detail (even when commercial tables are excluded), it must be borne in mind
that his descriptions are too short and general to be of great value, that more
than a third of his accounts are compiled from sources other than the original
works, and that he has made no attempt to do more than roughly classify the
works (not the tables) ; in fact a more detailed description or classification was
excluded by the plan of his article, which notwithstanding gives a great deal
of information in a very small space.

Art. 12. By an oversight (which was not discovered till it was too late to
remedy it) we have excluded from the Report traverse tallies, viz. Difference-
of-latitude and Departure tables, which under the head of multiples of sines
and cosines ought to have been noticed. Such tables are of general use in

* It is requested that communications may be addressed to Mr. J. W. L. Grlaisher,
Trinify College, Cambridge.

ON MATHEMATICAL TABLES. 13

all mathematics, as thoy are in reality merely tables for the solution of right-
angled triangles ; we have noticed one such table (Massalotjp, § 3, art. 10),
which was constructed for mining- (not nautical) purposes.

"We hope to repair the omission by appending a separate list of traverse
tables to a future Report.

Art. 13. A very important incidental gain that it was hoped would be
afforded by the present Report, was the opportunity of correcting errors in loga-
rithmic and other tables by giving references to the places in which errata-hsta
had been published. In the introductions or prefaces to works containing
tables, it is usual to give a list of the errors that have been found during
their preparation in previous tables ; and as few possessors of a work can be
acquainted with the publications that have appeared subsequently, it was
thought that by referring, under each title, to the works or periodicals in
which lists of errata in it had appeared, an important service would be rendered.
It was soon evident, however, that it was impossible to deal adequately with the
subject of errors in this manner. Many of the important collections have
been through very numerous editions ; and it was not always stated in which
editions the errors were found; and when the edition was stated, it was
doubtful (without examination) whether the errata-list in question had come
under the eye of the editor, and the errors been corrected in subsequent
editions, or not. In the case of stereotyped tables, successive tirages are more
and more accurate ; and in regard to collections of such tables published long
ago, as, for example, Callet (first published in 1783, though since reset), it seems
useless to waste space by giving references to the numerous errata-lists that
have been published, some of which must necessarily relate only to the earlier
tirages, and must have been corrected long ago. This is the case with all the
chief tables, and only in particular instances, when circumstances rendered it
probable that the errata-lists would be of use, have references been given to
them. As, however, this state of affairs is very unsatisfactory, it is hoped
that in a subsequent Report a complete list of errors in later editions of the
piost-used mathematical tables, still unsuperseded, may be given ; but it is ne-
cessary first to be satisfied that the errata given are not erroneous themselves.
Many of the chief modern lists of errata are noticed in this Report, and also
others that it seemed desirable to give references to at once ; but wo have
made no effort to deal with the matter in a complete manner. It is much to
to be regretted that it is not usual for editors of a new edition of a table to
give a list of the errors that occurred in former editions, and have been corrected
in that edition. It is only fair for the purchaser of a new edition of a work
to be informed wherein it differs from its predecessors ; but unfortunately the
object of the editor and publisher is to sell as many copies of the new edition,
not to render the old as valuable as the new. It is proper to add, however,
that usually, when tables are published by a mathematician for the advance-
ment of science, and not by a bookseller and editor for the sake of profit, an
exception is made to this rule, and errata are freely acknowledged. A remark
made by De Morgan with reference to mathematical books m general, viz.
that the absence of a list of errata means, not that there are no errors, but
merely that they have not been found out, is more appUeable to tables than
to any other class of work, in spite of the care usually bestowed on them ;
and an error in a table is far more fatal than an error m any other class of
work as there is no context (as far as the user is concerned) to show imme-
diately that the result taken from the table is erroneous. The subject of
errors will particularly occupy the attention of the Committee m a future
Report.

14 EEPOKT 1873.

Art. 14. The whole of the work requii-ed in the preparation of the Eeport
has been carefully performed ; and we believe that not many inaccuracies will
be found. Every work noticed, except only three or four, has been described
from actual inspection ; and the account has invariably been written with the
book before us. Every one, however, who has had any experience of biblio-
graphical work knows how impossible it is to be always accurate ; the work
has often to be performed in public libraries open only for a few hours in the
day, so that any one who has not an unlimited number of days at his command,
must sometimes work under pressure. Omissions are thus made, which, when
discovered during the revision six months afterwards, cannot be rectified
without great loss of time, even if it be remembered what library it was that
contained the work in question. The references from one part of the Eeport
to another wiU also, it is believed, be found correct j but as the whole plan
and arrangement have been altere.d in the course of the year over which the
preparation of the Eeport has lasted, it is possible that some of the old refer-i
ences may remain stUl uncorrected. If this should be found to be the case, not
much difficulty can ever be experienced in seeing what is meant with the aid
of the list of articles at the beginning of § 3, and the list of works in § 5 ;
also if any misprints (such as T. 11. for T. III. &c.) should escape notice iri.
the correction of the proofs, the reader will be enabled to correct these with-
out much waste of time. Lists of errata and corrections, should such be
needed, will be given in subsequent Eeports. Whenever we have made a
statement on some other authority than that of our own observation, we have
invariably stated it, though we are aware that we thus lay ourselves open to
the imputation of not having verified facts of the accuracy of which we might
have assured ourselves ; but, as De Morgan has observed, the possibility of
writing a history entirely from personal observation of the originals has not
yet been demonstrated,

§ 3. Separate Tables, arranged according to ttie nature of their contents ; with
Introductory Remarlcs on each of the Several hinds of Tables included iit
the present Report.

This section is divided into twehty-five articles, the subject matter of which
is as foUows : —
Art. 1. Multiplicatioii tables.

2. Tables of proportional parts.

3. Tables of quarter squares.

4. Tables of squares, cubes, square roots, and cube roots.

5. Tables of powers higher than cubes.

6. Tables for the expression of vulgar fractions as decimals.

7. Tables of reciprocals.

8. Tables of divisors (factor tables), and tables of pritnes.

9. Sexagesimal and sexcentenary tables.

10. Tables of natural trigonometrical functions.

11. Lengths (or longitudes) of circular arcs.

12. Tables for the expression of hours, minutes, .&c. as decimals of a

day, and for the conversion of time into space, and vice versd.

13. Tables of (Briggian) logarithms of numbers.

14. Tables of antilogarithms.

15. Tables of (Briggian) logarithmic trigonometrical functions.

16. Tablesofhyperboliclogarithms(viz.logarithmstobase2-71P28. . .).

17. Napierian logarithms (not to base 2-71828 . . .).

ON MATHEMATICAL TABLES. 15

Art. 18. Logistic and proportional logarithms.

19. Tables of Gaussian logarithms.

20. Tables to convert Briggian into hyperbolic logarithms, and wee versdt

21. Interpolation tables, '

22. Mensuration tables.

23. Dual logarithms.

24. Mathematical constants,

25. Miscellaneous tableSj figurate numbers, &c.

Art, 1, Muttvplication Tables.

The use of the multiplication table is so essential a part of the history of
Numeration and Arithmetic, that for information with regard to its introduce
tion and application -we must refer to Peacock's ' History of Arithmetic ' in
the ' Encyclopsedia MetropoHtana,' to De Morgan's 'Arithmetical Books'
(London, 1847), as well as to Heilbronner, Delambre, &c. (see § 2, art. 3),
to Leslie's ' Philosophy of Arithmetic,' and perhaps to Barlow's ' Theory of
Numbers' (London, 1811), in most of which references to other works will
be found. There is abundant evidence that, till comparatively recent titnes
(say the beginning of the eighteenth century), multiplication was regarded
as a most laborious operation ; this is testified not only indirectly by the very
simple examples given in old arithmetics, but explicitly by Decker in his
' Eerste Deel vande Nieuwe Telkonst ' (see Phil. Mag. Suppl. Number, Dec.
1872). The great popularity of Napier's bones, and the eagerness with
which they were received all over Europe, show how great an assistance the
simplest contrivance for reducing the labour of multiplications was considered
to be. It would be interesting to know how much of the multiplication
computers were in the habit of committing to tnemory, as the bones would
be no great help to any one who knew it as far as nine times nine. In this
Eeport, however, we are only concerned with extended multiplication tables
(viz. such as are to be used as tables, and were not intended to be committed
to memory). The earliest printed table of inultiplication we have seen re-
ferred to is Thomas Einck's ' Tabulae Multiplicationis et Divisionis, seorsim
etiam Monetae Danicse accommodatae,' Hafnise, 1604 (which title De Morgan
obtained from Prof. Werlauff, Eoyal Librarian at Copenhagen) ; but the
work, from its title, must have been rather a ready reckoner than a proper
scientific table. The earliest large table, which, strange to say, is still as exten-
sive as any (it has been equalled, but not surpassed by Ckbllb, 1864), is Heewaet
AH Hohenbttrg's 'Tabulse Arithmeticee vpoirOaijiaipeaecos Universales,' 1610,
described at length beloW. Of double-entry tables, Crellb's ' EeGbentafelUj'
1864, is the most useful, and the most used, for general purposes. The other
important tables are chiefly for multiplication by a single digit.

A multiplication table is usually of double entry, the two arguments being
the two factors ; and when so arranged, it is frequently called a " Pythagorean
Table." The great amount of room occupied by Pythagorean tables (no
table so arranged could extend to 1000 x 10,000, and be of practicable size)
has directed attention to modes of arrangement by which multiplication can
be performed by a table of single entry ; the most important of these aro
tables of quarter-squares, which are described in § 3, art. 3, where are also
added some remarks on multiplication tables of single entry. See also DiLinre,
described below.

It is almost unnecessary to add that, when not more than seven or ten
figures are required, multiplication can be performed at once by logarithms,
which (though not the best method for two factors when either a Pythagofean

16 REPOiiT — 1873.

or quarter-square table of suitable extent is at hand) have the advantage
that by their means any number of factors can be multiplied together at
once.

Getjson's table, 1798,is for multiplications of a somewhat diflferent kind from
the rest.

Crelle, in the introduction to hLs ' llechentafeln ' (1820), mentions a
work, ' Tables de Multiplication, a I'usagc de MM. les gcometres, de Mm. les
ingenieurs verifieateurs du Cadastre, etc' sec. edit. Paris, Chez Valace, 1812,
which he says extends to 500 x 500, and occupies 500 quarto pages ; while,
he adds, his own work, which is four times the extent, occupies only 1800
octavo pages. For the fuU titles of Picarte's ' Tables de Multiplication ' and
' Tableau Pithagorique,' see under Picabtb (1861), in § 3, art. 7.

Closely connected with multiplication tables are so-caUed Proportioiutl-parts
tables (described in the next article) ; and very frequently in the latter the
last figure is not contracted, so that by a mere change of the position of the
decimal point they become tables of multiples.

Herwart ab Hohenburg, 1610. Multiplication table, from 2 x 1 to
1000x1000. The thousand multiples of any one of the numbers are con-
tained on the same page, so that (as the number 1 is omitted) there are 999
pages of tables. By a strange oversight, the numbering begins with 1 on
the first page of the table instead of 2, so that the multiples of n are found
on page n— 1 : this is inconvenient, as the number of the page alone appears
on it, so that (say) to find a multiple of 898 we seek the page headed 897.
Each page contains 100 lines, numbered in the left-hand column 1, 2, 3, ... ;
and besides this column of arguments there are ten columns headed 0, 100,
. . . 900. The first figure of the multiplier is therefore found at the top of
"the column, and the last two in the left-hand column (on p. 3 it will be
noticed 200 and 300 are interchanged at the top of the columns). There
being more than 1000 pages of thick paper, the book, as Be Morgan has
observed, forms a folio of almost unique thickness. Also, as the pages con-
tain 100 lines, pretty well leaded, the size of the book is very large ; so that
Leslie (Philosophy of Arithmetic, 2nd edit. 1820, p. 246) was quite right in
calling it " a very ponderous folio." De Morgan says " the book is exces-
sively rare ; a copy sold by auction a few years ago was the only one we
ever saw."

Kiistner (' Geschichte,' t. iii. p. 8) quotes the remark of HeUbronner (who
gives the title of the work, • Hist. Math.' p. 801), " Docet in his tabulis sine
abaco multiplicationem atqne divisionem perficere," &c., and adds that Heil-
bronner could not have seen the work, or he would have described it ; he
remembers to have read that it was like a great multiplication table. The
title is given by Murhard, and marked with an asterisk to show that he had
seen a copy. Kogg gives the title very imperfectly ; and it is clear the work
has not been in his hands. There is a complete copy in the British Museum,
and a copy in the Graves Library; but the latter is imperfect, the pages
12-25, 120-145, and 468-517 having been lost, and their places supplied
with blank paper. On account of the rarity of the work, and the great in-
terest attaching to it from the time when it was published, we have thought
it worth whUe to give the title in full in § 5. The clearness of the type
and the extent of the table (which has not been surpassed, and only equalled
by Ceellb, 1864), taken in connexion with its early date (four years before
Napieb's ' Canon Mirificus '), give the work a peculiar interest. De Morgan
writes : — ■" it is tnily remarkable that while the difldculties of trigonometrical

ON MATHEMATICAL TABLES. 17

calculations were stimulating the invention of logaritlinis, they wore also
giving rise to this the earliest work of extended tabulated multiplication.
Herwart passes for the autlior ; but nothing indicates more than that the
manuscript was found in his possession." We have seen the statement that
while Napier solved triangles by logarithms, Herwart did so by prosthaphse-
resis, and others of the like kind, the inference being that Herwart invented
a method which has been superseded by logarithms; this (if the present
work is the source of the statement) is incorrect, Herwart's table being
merely useful in facilitating the multiplications required in the formulse.

There are in the British Museum three other works of Herwart ab Hohen-
burg : viz., ' Thesaums Hieroglyphicorum e museo Joannis Georgii Herwart
ab Hohenburg . . .' (Obi. fol. Munich ?, 1610 ?) ; ' Nov£e, verse et exacts ad cal-
culum . . . Chronologise c museo . . .' Small 4to, 1612; and 'Ludovicus Qnartus
Imperator defensus . . . ab Joanne Georgio Herwarto' &c. 4to. Munich, 1618
(the middle one of which is given in Lalande's Bib. Ast.). "We have looked
at these three books in the hope that some mention might be made in them
of the table, or some information given about Herwart's Museum ; but they
appear to contain nothing of the kind. We have seen also the titles of several
other works of Herwart's, and references to where particulars of his life are
to be found ; so that, considering the attention so large a work as his table
must have received from contemporary mathematicians, we still have hopes
of being able to bring to light some information with regard to its calculator,
his objects, &c.

It should be stated that Herwart ab Hohenburg is spoken of quite as fre-
quently by the name of Hohenburg as by that of Herwart.

The author of the anonymous table (1793) described below, states that
many errors were found in Hbkwaet, and that Schiiblor (whose table we have
not seen) was much more correct. .

Riley, 1775. The first nine multiples of aU numbers from 1 to 5280.
The multiples of the same number are placed one under the other, the factors
1, 2 , , , 9 being three times repeated on the page, which contains ten columns
of results and twenty-seven lines.

The preface is signed Geo. Eiloy and T. O'B. Macmahon. There is an ad-
vertisement of Eiley's " historical playing-cards" &c. at the end, and of several
works by Macmahon. On the relation of this book to another, " printed for
J. Plummer" (anonymous) in the same year, see De Morgan.

Anonymous, 1793.' Multii)lication table exhibiting products from 2 x 13
to 100 X 1000, arranged so that there are 100 multiples (in two columns) of
four numbers on each page, which therefore contains eight columns.

Gruson, 1798. The first part of this book contains a number of tables,
the description of any one of which will explain the arrangement. Take the
table 86 : it has ten columns, headed 0, 1, 2, . . . , 9 (as have all the other
tables), and 36 lines, numbered 0, 1, 2, . . . , 35; we find in column 6 and
line 21 (say) 237=6 x 36+21. The use of the table is as follows : — suppose
it required to find the number of inches in 6 yards 21 inches ; 36 in. =1 yd.,
we find table 36, column 6, line 21, and have the result given in inches.
There are tables for all numbers from 1 to 100, and for primes from 100 to
400, the number of lines in each table being equal to .the number of the
table. The use of the tables in performing ordinary divisions and multipli-
cations when there are four or more figures in the divisor or dividend, &c. is
fuUy explained by the author in the introduction. When used for division,
the table gives the quotient and the remainder.

There is also given a table of aU simple divisions of numbers (not divisible
1873. ,

18 KEPORT— 1873.

by 2, 3, or 5) to 10,500. A short and grandiloquent dedication to the
French Institute is prefixed.

Eogg gives also a German title, ' Piuacothek, oder Sammlung allgemein-
niitzlicher Tafeln fiir Jedermann ' &c.

Gruson, 1799. A table of products to 9 X 10,000. The pages, which
are very large (containing 125 lines), are divided into two by a vertical lino,
each half page containing ten columns, giving the numbers and their first
nine multiples : the first half of the first page thus ends at 9 x 124, the
second half at 9 x 249 ; and there are 1992 tabular results to the page. The
table has only one tenth of the range of Beexschneidee's ; but the result is
given at once ; however, the large size of the page (almost, if not quite, the
largest we have seen for a table) is a great disadvantage. There are two
pages of explanation &c.

The title describes the table as extending to 100,000, the above being only
the first part. We do not know whether any more was published, but think
probably not. Eogg mentions no more. At the end of the introduction
three errors occurring in some copies are given.

Martin, 1801. This is a large collection of tables on money-changing,
rentes, weights and measures, &o. The only part of the book that needs
notice here is Chapter XI., which contains a multiplication table giving the
first nine multiples of the numbers from 101 to 1052 (19 pp.).

Dilling, 1826. In the use of a table of logarithms to multiply numbers
together, the logarithms used are of no value in themselves, being got rid of
before the final result. If, therefore, letters a,h,e,4.. be used instead, we
have no occasion to-know the values of any one of them, but only the way in
which they are related to one another. The present table is constructed for
numbers up to 1000 on this principle; within this range there are about 170
primes, the logarithms of which have to be denoted by separate symbols,
a, b, ..,, z, a^, 6j, . . . , &e. ; the powers of 2 are denoted by numbers ; thus
log (2*)=2, log (2^)=3, &c. ; and the logarithms of any number to 1000 can
be easily expressed in not more than four terms; thus log 84=2+a+c.
There is also a table of antilogarithms arranged according to the last letter
involved; thus log 21=a+c, log 15=a+b, the sum =!=2a+h-\-c; and
entering the antilogarithmic table at e, we find 315 the product. We can
thus only multiply numbers whose product is less than 1000 ; and a table of
products of the same size would certainly have been more useful. The table
can of course be used for division, square roots, Sec, but only if the result is
integral, so that it is little more than a matter of curiosity. This table was
intended, however, only as a specimen, to be followed by a larger one to
10,000. We believe the continuation was not published ; and Eogg refers to
no other work of Pilling.

The work, although nominally a table of logarithms, is included in this
article, as it is really a multiplication table. It is the only table we have met
with involving a principle which at one time would have been of value with
respect to multiplication, viz. to resolve the numbers into their prime fectora,
and multiply them by adding their factors. Thus 21=3 x 7, 15=3 x 5, and
their product 315=3^ X 5 X 7 ; if therefore we had a table giving the prime
factors of aU numbers from 1 to 1000, arranged in order, and another table
of like extent giving the numbers corresponding to the same products of
factors, arranged with the largest factor first, and the others in descending
order, so as to facilitate the entry, we could perform multiplication (where
the product does not exceed 1000) by addition only. In the construction of
sueh a table it would soon be found convenient to replace the two and three

ON MATHEMATICAL TABLES. 19

figure primes by letters, to save room, and, in fact, to use letters throiieh-
out-and further to ^simplify the printing by writing <x* as 4a, &c., which
would do equally well; wo then have Duxinq's tables^ which have not the
smallest connexion with logarithms. Such a table might once have been
tound useful; but the slightest consideration shows that (except as a factor
table) It would be all but valueless now. The space a large table of the kind
would occupy, the impossibility of arranging the antifactor table so as to,
admit ot easy entry, and the great convenience of existing tables (both
iyttiagorean and logarithmic) are alone sufficient to prove this.

Crelle, 1836. This table occupies 1000 pages, and gives the product of
a number of seven figures by 1, 2, ... , 9, by a double operation, very much
m the same manner as BrbiscAneidbe's does for a number of five : viz , each '
page IS divided into two tables; thus, to multiply 93BgS'7 by 7, we turn to
page 825, and enter the right-hand table at line 77, d^ilin 7, where we find
77339; we then enter the left-hand table on the same page, at line 93,
column 7, and find 656, so that the product required is 65677339. "We tbink
for numbers seven figures long the table efi'ects a considerable saving of time,
as it is as easy to use as Beetschneideb's for five figures. It would take some
little practice to use the table rapidly in all cases, as qf course the mode of
entry, (fee. must be varied according as the number consists of seven, sis,
five, &c. figures ; but the value of a table is measured not by the trouble
required to learn to use it, but by the time saved by means of it after the
computer has learnt its use.

Bretschneider, 1841. This table is for the multiplication of any
number up to 100,000 by a single digit. On each page there are two tables,
the upper of which occupies ten lilies, and the lower fifty. An example will
show the method of using the table, Suppose' it required to multiply
56878 by 7, then the table is entered on the page headed 6800 (the headings
run from to 99, with two ciphers added to each). Facing 78 in the lower
table we find *146 ; and in the upper table facing 568, in the column for 7,
we find 397;- the product required is therefore 398146, the third figure
being increased because the 146 was marked by an asterisk. The arguments
in the upper table, on the page headed 6800, are 68,168,268 ... 968 (twice
repeated for the two cases when succeeding numbers are less and greater
than 50), eind also 1, 2 ... 9, as the table is of double entry.

The arrangement of the table is thus very ingenious ; but, as De Morgan
has remarked, multiplication by a single digit is so simple an operation that
it is questionable how far a table is serviceable when its use requires three
distinct points to be attended to.

The introduction (10 pages) gives a complete explanation of how the table
can be used when the number of figures is greater than five. Haying made
some use of the table for this purpose, we dp, not think any time is saved by
it ; at all events, not until the computer has had much practice in using it.

Crelle, 1864. This magnificent table gives products up to 1000 x 1000,
arranged in a most convenient and elegant manner, one consequence of which
is that all the multiples of any number appear on the same page. It is also
very easy to get used to the arrangement of the table, which is as useful for-
divisions as multiplications. It can be used for multiplying numbers which"
contain more than three figures, by performing the operation, three figures
at a time ; but it requires some practice to do this readily ; and a similar
remark applies to the extraction of square roots.

There is one great inconvenience that every computer must feel in using
the work, viz. thftt the multiples of numbers ending; in are omitted, so that,

c2

20 REPORT— 1873.

for example, we pass from 39 to 41. It is quite true that the columns for
40 are the same as those for 4 with the addition of a ; but the awkward-
ness of turning to opposite ends of the book for (say) 889 and 890, and then
having to add a to the latter, is very great. It is a pity that a desire to
save a few pages should have been allowed to impair the utility (and it does
80 most seriously) of so fine a table. The matter is referred to in the
preface, where it is said that Crelle, " after mature reflection," decided to
omit these numbers.

The original edition was published in 1820, and consisted of two thick
octavo volumes, the first proceeding as far as 500 x 1000, and the second
completing the table to 1000 x 1000. The incourenience referred to above
is felt more strongly in this than in the one-volume edition, as frequently the
numbers ending in have to be sought in a different volume from the others.
Both editions are, we believe, very accurate. There are 3 pp. of errata
(pp. xvii-xix) at the beginning of the edition of 1820. De Morgan gives
1857 as the date of Bremiker's reprint, and says he has heard that other
copies bear the date 1859, and have no editor's name.

Itaundy, 1865. The first nine multiples of aU numbers from 1 to 100,000, .
given by a double arrangement : viz., if it is required to multiply 15395 by 8,
we enter the table on p. 4 (as 395 is intermediate to 300 and 400) at 15,
and in column 8 find 122 ; we enter another table on the same page at 395,
and in column 8 find 160; the product is therefore 123160. We take this
number instead of 122160 because in the column headed 8, first used, there
appears the note [375]*, the meaning of which is that if the last three figures
of the number exceed 375 (they are 395 in the above example) the third
figure is to be increased by unity. The table is thus seen to be the same in
pi-inciple as Bretschnbidee, but not quite so convenient. There are the same
objections to this as to the latter table. The present table occupies 10 pp.
4to, and Beeischneidee's 99 pp. 8vo.

Mr. Laundy remarks in his preface that Csblle's ' Erleichterungs-Tafel,'
1836, although one hundred times as large as his, " must not be estimated as
presenting advantages proportionate to its vast difference of extent." In this
we scarcely agree ; for it is only when the numbers are six or seven figures
long that one begins to feel the advantages of a table for so simple an operation
as multiplication by a single digit, and Ceeile's table would not take much
longer to use than the present.

The following is a list of references to § 4 : —

MuMplication Tables.— T)a-asoTS, 1747, T. XXXVIII. to 9 x 9999 ; Hotton,
1781 [T. I.] to 100 X 1000 ; Callet, 1853 [T. VIII.] ; Scheon, 1860, T. III. ;
Paekhtost, 1871, T. XXVI., XXXIII., and XXXIV.; see also Leslie,
1820, § 3, art. 3, and Wtjcheeek, 1796, T. II. (§ 3, art. 6.)

Art. 2. Tables of Proportional Parts.
By a table of the proportional parts of any number x is usually under-
stood, a table giving J^a;, ^-^x,. ..-f^x true to the nearest unit. Of course
the assumption of 10 as a divisor is conventional, and any table giving

X 2x (a 1^.17

-, — , . . . =^ — would equally be called a proportional-part table. Ordi-

. nary proportional-part tables (viz. in which a =10) are given at the sides of
the pages in all good seven-figure tables of logarithms that extend from
10,000 to 100,000. The difference between consecutive logarithms at the
commencement of the tables (viz. at 10,000) is 434, and at the end is there-
fore 43 ; so that a seven-figure table of the above extent gives the proportional

ON MATHEMATICAL TABLES. 21

parts of all numbers from 43 to 434 (note that near the commencement of
the table, viz. from diff. 434 to diff. 346, the proportional parts are only
given for every, other difference in some tables ; whether a table gives the
proportional parts of all the differences or not is generally noted in § 4).
Several seven-figure tables extend to 108,000 ; and for the last 8000 the dif-
ferences decrease from 434 to 403. Tables in which a=60 often accompany
canons of trigonometrical functions that give the results for every minute, for
convenience of interpolating for seconds; such must be sought from the
descriptions of trigonometrical tables in § 3, arts. 10 and 15, and in § 4 ;
we have also seen tables for which a =30, where the functions are tabulated
for every two minutes or two seconds.

There are several tables to which proportional parts of the differences to
hundredths (viz. in which a =100) are attached, e.g. Geat (§ 3, art. 19),
Piiipowssi (§ 4), and Pineio (§ 3, art. 13) ; but the ranges of the differences
are generally so small that it is not worth while giving references. In
PiNEio, for instance, the range of the differences is only from 4295 to 4343
(in this work multiples are given, the last two figures being separated by a
comma).

The only separate table of proportional parts, properly so called, that we
have seen, is

Bremiker, 1843 (' Tafel der Proportionaltheile'). Proportional parts to
hundredths (viz. multiples from 1 to 100, with the last figure omittedj and
the last but one corrected) of all numbers from 70 to 699. A very useful
table, chiefly intended for use in interpolating for the sixth and seventh figures
in logarithmic calculations.

T. III. of ScHEON (§ 4) (which is there called an Interpolation Table) is a
large table of proportional parts.

It is to be noticed that all multiplication tables are, or rather can be used
as proportional-part tables. A table of multiples, with the last figure omitted,
and the last but one corrected (which can be done at sight), is a proportional-
part table to tenths ; and if. the last two figures are omitted, and the last
remaiuing figure corrected, to hundredths (see therefore § 3, arts. 1 and 3).

It is proper here to allude to slide-rules and other mechanical appliances
for working proportions &c. A card intended to do the work of a very large
slide-rule is described in § 4 (Evbkbtt) ; and some information and references
about slide-rules of different shapes will be found in a paper " On a New
Proportion Table," by Prof. Everett, in the Phil. Mag. for Nov. 1866.

The following are references to works described in § 4 : — ,

Tables of Proportional Parts.— Sir J. Mooee, 1681 [T. II.]; Dttcom, 1820,
T. XX.; LxNN, 1827, T. Z; Callbi, 1853 [T. VIII.]; Scheon, 1860,
T. III.

Art. 3. Tables of Quarter Squares.

Tables of quarter squares have for their object to facilitate the performance
of multiplications ; and the principle on which their utility depends is con-
tained in the formula

a6=|(a+J)=-|(a-6)^

so that with such a table to multiply two numbers we subtract the quarter
square of the difference from that of their sum ; the multiplication is there-
fore replaced by an addition, a subtraction, two single entries of the tables,
and a final subtraction — a very considerable saving if the numbers be high.
The work is more than with a product table, where a double entry gives the
result at once; but the quarter squares occupy much less space, and can

22 EEPOKT — 1873.

therefore be tabulated to a much greater extent without inoonvenienoe. In
tables of quarter squa,re8 the fraction | -whioh occurs when the number is
odd is invariably lefb out ; this gives rise to no difficulty, as the sum and
difference of two numbers must be both odd or both even.

A product can, of course, be obtained by logarithms with about the same
facility as by a table of quarter squares ; but the latter is preferable when all
the figures of the result are required.

Ltjdolf, 1690 (see § 3, art. 4), in the preface to his ' Tetragonometria,'
explains the method of quarter squares completely, and shows how his table
is to be used for the purposes of multiplication. The earliest table of quarter
squares De Morgan had heard of was Voisin, 1817 ; but Centneeschweh (see
below) refers to one by Biirger of the same date, the full title of which we
have quoted from Eogg.

Cbbilb, in the preface to the first edition of his ' Eeohentafeln ' (1820,
p. XV.), speaks of " Quadrat-Tafeln nach Laplace und Gergonne, mittelst
welcher sich Produote findon lassen," &c. The allusion to Laplace doubtless
refers to the memoir in the ' Journal Polytechnique,' noticed fui-ther on in
this article ; but we cannot give the reference to Gergonne.
- The largest table of quarter squares that has been constructed is that
published by the late Mr. Lattndy, whioh extends as far as the quarter
square of 100,000 ; it would be desirable, however, to have a table of double
this extent (viz. to 200,000), which would perform at once multiplications of
five figures by iive figures (Mr. Laundy's table is only dii'ectly available
when the sum of the numbers to be multiplied is also of five figures). The
late General Shortrede constructed such a table, we believe, in India, but
unfortunately abandoned the idea of publishing it on his return to England,
where he found so much of the field already covered by Laundy's tables.
De Morgan, writing when it was anticipated that Shortrede's table would be
published, suggested that it would be convenient that the second haK should
appear first ; and we should much like to see the piibUoation of a quarter-
square table of the numbers from 100,000 to 200,000. '

Sir. Lattndy, in the preface to his ' Table of Quarter Squares ' (p. vi), says
that Galbraith, in his ' General Tables,' 2nd edit. 1836, whioh were intended
as -a supplement to the Second edition of his ' Mathematical and Astronomical
Tables,' gives a table (T. xxxiv.) of quarter squares of numbers from 1 to
8149. This book is neither in the British Museum nor the Cambridge Uni-
versity. Library. The second edition of his ' Mathematical and Astronomical
Tables ' (1834) contains no such table. There is, however, no doubt about
the existence of the work, as the Babbage Catalogue contains the title
" Galbraith, W., New and concise General Tables foi computing the Obliquity
of the Ecliptic, &c. Edinburgh, 1836."

In 1854, Prof. Sylyester having seen a paper in Gfergonne in which the
method was referred to, and not being aware that tables of quarter squares
for facilitating multiplications had been published, suggested the calculation
of Such tables, in -two pefpers — " Note on a Formula by aid of which, and of a
tabic of single entry, the continued product of any set of numbers . . . may be
effected by additions and subtractions only without the use of Logarithms "
(Philosophical Magazine, 8. 4. vol. vii. p. 430), arid " On Multiplication by
aid of a Table of Single Entry " (Assurance Magazine, vol. iv. p. 236). Both
these papers were probably written together ; but there is added to the former
a postscript, in which reference is made to Voisin and Shortrede's manuscript.
Prof. Sylvester gives a generalization of the formula for ab as the difference
of two squaresj in which the product a^ a^ . . . ff„ is expressed as the sum of

ON MATHEMATICAL TABLES. 23

Jith powers of a^, a^, . . . «„, connected by additive or subtraetive signs. For
the product of three quantities the formula is

And at the end of the 'Philosophical-Magazine' paper, Prof. Sylvester has
added some remarks on how a table to give triple products should be
arranged.

At the end of a memoir, " Sur divers points d'Analyse," Laplace has given
a section " Sur la Eeduction des Fonctions en Tables " (Journal de I'Ecole
Polytechnique, Cab. xv. t. viii. pp. 258-265, 1809), in which he has briefly
discussed the question of multiplication by a table of single entry. His
analysis leads him to the method of logarithms, quarter squares, and also to the
formula sina sin5=|{co3(a-6)--co8(ffl + 6)}, by which multiplication can
be performed by means of a table of sines and cosines. On this he remarks,
" Cette mani^re ingenieuse de faire servir des tables de sinus h, la multiplication
des nombres, fut imaginee et employee un siecle environ avant I'invention
des logarithmes."

It is worth notice that the quarter-square formula is deduced at once from
sin a sin 6 =§ { cos (a — 6) — cos (a + 6)}, by expanding the trigonometrical func-
tions and equating the terms of two dimensions ; similarly from sin a sin 5
sin c=|{sin (a+c-6)+sin(ffl + 5-c) + sin(6+c-a)-sin (a + 6+ c)}, by
equating the terms of three dimensions, we obtain a5c=-^{(a+64- c)'— &c.f,
as written down above, and so on, the general law being easily seen. We
may remark that there is an important distinction between the trigonometrical
formulae and the algebraical deductions from them, viz. that by the latter to
multiply two factors we require a table of squares, to multiply three a table
of cubes, and so on j i, e. each different number of factors requires a sepa-
rate table; while one and the same table of sines and cosines will serve to
multiply any number of factors. This latter property is shared by tables of
logarithms of numbers, the use of which is of course in every way preferable ;
still it is interesting to note the inferiority that theoretically attaches to the
algebraical compared with the trigonometrical formulae. Other remarks on the
subject of multiplication by tables are to be found in § 3, art. 1,

It is almost unnecessary to remark that a table of squares may be used
instead of one of quarter squares if the semisum and semidifference of the
numbers to be multiplied be taken as factors. Tables of squares and cubes
are described in the next section.

*Voisin, 1817. Quarter squares of numbers from unity to 20,000. "We
have taken the title from the introduction to Mr. Lathstdt's ' Quarter Squares'
(1856). De Morgan also so describes the work. We have seen no copy ; but
there is one in the Graves Library, although We were unable to find it : it
will be described from inspection in the supplement to this Eeport.

Leslie, 1820. On pp. 249-256 there is a table of quarter squares of
-numbers from 1 to 2000, reprinted from Voisar, 1817, whose work Leslie
met with at Paris in 1819. There is also given, facing p. 208, a large folding
sheet, containing an enlarged multiplication table, exhibiting products from
11 X 11 to 99 X 99, the table being of triangular form. There are also, on
the same sheet, two smaller tables, the first giving squares, cubes, square
. roots (to seven places), cube roots (to six places), and reciprocals (to seven
places) of numbers from 1 to 100, and the second being a small multiplication
table from 2x 2 to 25 x 25. In the first edition (1817, pp. 240) the quarter-
square table does not appear ; and in the folding sheet (which follows the

24 REroBT — J 873.

preface) the smaller multiplication table is not added ; squares and cubes only
are given in the other small table.

Centnerschwer, 1825. [T. I.] A table of quarter squares to 20,000 ; viz.

V, is tabulated from .r=l to i);=20,000, the fraction |, which occurs when

X is odd, being omitted. The last two figures of the quarter square, which
only depend on the last two figures of the number, are given once for all
on two slips bound up to face pp. 2 & 41.

Full rules are given as to how to use the table as a table of squares ; and
three small tables are added, by means of which the square of any number
of Jive figures can be found tolerably easily. The arguments are printed
in red.

[T. II.] Square roots of numbers from 1 to 1000 to six places.

There is a long and full introduction prefixed.

In his preface Centnerschwer states that after his work was in the press,
he received from Crelle a table, by J". A. P. Biirger, entitled " Tafeln zur
Erleichterung in Eechnungen," Karlsruhe, 1817, in which the author claims
to be inventor of the method, while Centnerschwer states it was known to
Ltjdolf (1690), and even Euclid. That Ludolp was the inventor of the
method is true ; and there is attached to his work a table of squares to
100,000 (see Ltjdolf, § 3, art. 4).

The full title of Biirger's work, which we have not been successful in ob-
taining a sight of, is (after Eogg) as follows : — " Tafeln zur Erleichterung in
Eechnungen fUr den allgemeinen Gebrauch eingerichtet. Deren ausserst ein-
fach gegebene Eegelnj nach welchen man das Product zweier Zahlen ohne Mul-
tiplication finden, auch sie sehr vortheilhaft bei Ausziehung der Quadrat- und
Cubicwurzel anwenden kann, sich auf den binomLschen Lehrsatz griinden.
Nebst Anhang iiber meine im vorigen Jahr erschienene Paralleltheorie.
Carlsruhe, 1817. 4to." The book last referred to was entitled " Vollstandige
Theorie der Parallellinien &c. Carlsruhe, 1817 ; 2nd edit* 1821," as given
by Eogg under Elementar-Geometrie.

Merpautj 1832. The premiire partte gives the arithnome (i. e. quarter
square) of all numbers from 1 to 40,000, so arranged that the first three
figures of the argument are sought at the head of the table, the fourth figure
at the head of one of the vertical columns, in which, in the line with the final
(fifth) figure in the left-hand column, is given the quarter square required.
The quarter squares are printed in groups of three figures, the second group
being under the first, &c. A specimen of this table is given by LAUNDr
(1856, p. v of his Introduction).

The deiixUme partie gives the reciprocals of all numbers from 1 to 10,000
to nine figures.

The author seems not to have been aware of the existence of any of the
previous works on the subject of quarter squares.

Lauudy, 1856. Quarter squares of all numbers from unity to 100,000
the fraction |, which occurs when the number is odd, being, as usual, omitted.
The arrangement is as in a seven-figure logarithm table ; viz. the first four
figures are found in the left-hand column, and the fifth in the top row ; the
three or four figures common to the block of figures are also separated as in
logarithmic tables, and the change in the fourth or fifth figure is denoted by
an asterisk prefixed to all the quarter squares affected : at the extreme left
of each page is a column of corresponding degrees, minutes, and seconds
(thus, corresponding to 43510 we have 12° 5' 10"=43510"). At the bottom
of the page are diflferences (contracted by the omission of the last two figures)

ON MATHEMATICAL TABLES. 35

and proportioual parts. The figures are very clear ; aad tliore is a full intro-
duction, with explanations of the use, &c. of the tables.

Mr. Laundy was induced to construct his table by Prof. Sylvester's paper
in vol. iv. of the 'Assurance Magazine,' referred to above ; and a description
of the mode of construction &c. of the table (most of which js also incor-
porated in the introduction to it) is given in vol. vi. of the ' Assurance
Magazine.'

Art. 4. Tables of Squares, Cubes, Square roots, and Cube roots.

Tables of squares (or square roots of square numbers) are of nearly as
great antiquity as multiplication tables, and would, we think, be found to be
rather common in early manuscripts on arithmetic. They are, as a rule, but
slightly noticed in histories of the subject (see references in § 3, art. 1), partly
because the latter are very meagre, and very many manuscripts remain still
unexamined, and partly because it is rather the province of a history to de-
scribe the improvement of processes. The perfection of the methods of ex-
tracting the square root of numbers not complete squares, however, occupies
a conspicuous place.

In the MSS. Grg. ii. 33 of the Cambridge University Library, are two frag-
ments, one of Theodorus Meletiniotes,the second of Isaac Argyrus (both much of
the same date, time of John Palaeologus, 1360) (concerning the first, see Viu'
cent, Manuscrit de la Bibhoth^que linperiale, xix.pt. 2. p. 6). The fragment
is a portion of the first book, and contains rules and small tables for multi-
plication, fractional computation &c.

The tract of Isaac Argyrus is entitled " tov 'Apyvpov eipeais tiSv rerpayoi-
riKiSv TrXevpiSv TiSy /ii) pijriiii' apid/xiSy.

At the end there is a table of the square roots of all integral numbers from
1 to 120, in sexagesimal notation. The table is prepared as if for three
places of sexagesimals ; but usually two only are perfect. Errors (probably
due to the copyist) are frequent. Before the table is a description of the
method of its use, including an explanation of the method of proportional
parts.

De Morgan speaks of two early (printed) tables in Pacioli's ' Summa/
1494, and by Cosmo Bartoli, 1564, extending respectively to the squares of
100 and 661. The tables which we have examined are described below; but
there are several of some extent, which De Morgan refers to, that we have not
seen, viz. : — Guldinus, 1635, squares and cubes to those of 10,000 ; W. Hunt,
1687, squares to that of 10,000 ; and J. P. Biichner's ' Tabula Eadicum,'
Nuremberg, 1701, which gives squares and cubes up to that of 12,000 (full
title given in Kogg). Lambeki (Introd. ad Suppl. &c. 1798) says that
Biichner's table is "plena errorum." Eogg gives the title " Bbbert, K. W.,
Tafeln der Quadratzahlen aller natiirlichen Zahlen von 1-25,200 ; der Kubik-
zahlen von 1-1200; der Quadrat- u. Cubicwurzeln von 1-1000. Neu berechnet,
Leipzig, 1812 ;" and the title occurs in the Eoy. Soc. Lib. Cat. (though the
book is not to be found in the Library). De Morgan mentions " Schiert,
' Tafeln,' &c. Eohn om Eheim, 1827," as giving squares to 10,000, which is
no doubt a misprint for " Schiereck, J. F., Tafeln aller Quadrate von 1 bis
10,000. 4to. Kiiln am Ehein, 1827," which occurs in the Babbage Catalogue,
and also in Eogg. Erom the title of another work of Schiereck's given in
the former catalogue, it appears that the table of squares also appeared as an
appendix to his :• Hahdbuch fiir Geometer/ published in the same year.

De MoEeAB- speaks of Ludolp's . ' Tetragonometria,' 1690, which gives
squares up to that of .100,000," as being the largest in existence, and very

26 REPOKT — 1873.

little known." This is true ; but Kulik, 1848, la of the same extent, and
also gives cvbes up to that of 100,000, thus giving the largest table of squares,
and by far the largest table of cubes in the same work, and in a compact and
convenient form : of this work also it may be said that it is very little known.

Htttton, 1781 (§ 4), gives squares to that of 25,400, and cubes to that of
10,000 ; but for most purposes Bablow (stereo. 1840), which gives squares,
cubes, and square roots and cube roots (and reciprocals) of numbers to 1000,
and is very accurate, is the best. We have not seen any square-root or cube-
root table of greater extent.

Extensive tables of quarter squares have been published, which are de-
scribed in § 3, art. 3 ; and some tables of squares, as FaA de BKirifo, were
constructed with the view of being used in applying the method of least
squares.

It is scarcely necessary to remark that logarithms find one of the most
valuable applications in the extraction of roots. Multiplications &c. can be
performed generally without their aid with a little more trouble ; for finding
square and cube roots they are extremely useful ; but for the extraction of
higher roots there exists no other method admitting of convenient application.

Maginus, 1692. The ' Tabula Tetragonica ' is introduced by the words
■" sequitur tabula numerorum quadratorum cum suis radieibus nunc primum
ab auctore supputata, ac in lucem ssdita," and occupies leaves 41-64. It
•gives the squares of all numbers from 1 to 100,100. We have seen the
' Tabula Tetragonica ' quoted as an independent work ; and De Morgan says
■that it was published separately, with headings and explanations in Italian
instead of Latin. In the copy before us Tavola is misprinted for Tahvla on
pp. 41 and 43 back (only the leaves are numbered).

The work contains sines, tangents, and secants also.

Magini was, we suppose, the vernacular name of the author, and Maginus
the same Latinized. We have somewhere seen Magini and Maginus spoken
-of as if they were different persons.

Alstedius, 1649. In part 3. pp. 254-260, Alsted gives a table of squares
and cubes of numbers from 1 to 1000. Alsted's is the first CyclopSedia, in
the sense that we now understand the word.

[Moore, Sir Jonas, 1650?] Squares and cubes of numbers from 1 to
1000, fourth powers from 1 to 300, fifth and sixth powers from 1 to 200.

In the book before us (Brit. Mus.) this tract (which has a separate pagina-
tion) is bound up at the end, after Moore's 'Arithmetiek (and Algebra),
Contemplationes Geometriose, and Conical Sections.' De Morgan says that
power tables, exactly the same as these, were given in Jonas Moore's ' Arith-
metic ' of 1650, and reprinted in the edition of 1660 ; so that probably the
tract noticed here usually formed part of the 'Arithmetiek.'

[Pell], 1672. Squares of numbers from 1 to 10,000 (pp. 29), This is
followed by the 6 one-figure endings, the 22 two-figure endings, the 159
three-figure endings, and the 1044 four-figure endings, which square numbers
admit of. They are given at length, and also in a synoptical form. The last
page in the Eoy. Soo, copy is signed John Pell. (In the Eoyal Society's Li-
brary Catalogue this table is entered under Fell, the signature at the end in
the Society's copy having been struck out so as to render the first letter
uncertain.)

In the Brit. Mus. is a copy without any name (so that perhaps PeU's name
was supplied in the Roy. Soo. copy only in manuscript). ' Dr. Pell's Tables,'
Tiowever, is written in it, and no doubt can exist about its authorship.

ON MATHEMATICAL TABLES. 27

Ludolf, 1690. Squares of all numbers from unity to 100,000, arranged
in columns, so that the first three or four figures of the root are to be found at
the top of the column, while the final ones are given in the left-hand column of
the_ page. The table is -Well printed and clear, and, except Ktoik, 1848,
which is of the same extent, is the largest table of squares that has been
published, and occupies about 420 pages. Some errata in it are given at
the end of the introduction (150 pp. in length), in which aU possible uses
of the table are explained.

Lambert (Introd. ad Supplementa, 1798) speaks of the numbers in the
table as " satis accurati." In chapter v. (pp. 48-86) (' De Tabularum usu
sen Praxi circa Multiplicationem et Divisionem ') the use of the table as one
of quarter squares (see § 3, art. 3) is fully explained ; as squares are given
in the table, the sum and difference have to be divided by 2. Eules and
examples are also added as to how to proceed when the semisum exceeds the
limits of the table by any amount ; and the processes &c. are explained with
Such fulness as to prove that aU the credit of first perceiving the utility of
the method and calculating the necessary table is due to Ludolf.

The work is said to be very scarce ; but we have seen several copies ; there
is one in the Library of Trinity College, Cambridge, and another in the
Graves Library.

Heilbronner (under Heewakt ab HoffENUijEG) mentions Ludolf (Hist. Math.
p. 827), and (referring doubtless to the method of quarter squares) Sd,ys that
he invented a method of performing multiplications and divisions without the
Pythagorsean abacus, " quae prolixe ab Illustr. Wolflo in seinen Anfangs-
Griinden et suis Elementis Matheseos exponitur."

Seguin, 1786. At the end of the book is given a table of the squares and
cubes of numbers from unity to 10,000. The figures have heads and tails,
and are very clear. De Morgan states that the table was reprinted at about
the beginning of the century, and that it was this table which convinced him
of the superiority of the numerals with heads and tails, and led him in the
reprint of Lalande's table, 1839, to adopt this figure— an example which has
since been very frequently followed.

As De Morgan does not appear to have seen it, it is possible that the ori-
ginal table was not reprinted, but only published separately, as the figures in
the table attached to Seguin answer De Morgan's description very well.

Barlow's tables (the stereotyped edition of 1840). Squares, cubes, square
roots, cube roots, and reciprocals to 10,000. The square roots and cube roots
are to seven places, and the reciprocals to seven significant figures, viz. nine
places to 1000, and above this ten. The work is a reprint of the more im-
portant tables in Baklow, 1814 (described in § 4) ; it was suggested by De
Morgan, who wrote the preface (2 pp.), and edited by Mr. Farley, of the
Nautical-Almanac Office, who also examined carefully Barlow's tables. A
list of ninety errors found in the latter is given on the page following the
preface. This reprint is, we believe, very nearly, if not quite, free from
error ; it is clearly printed and much used. "We have also an edition, 1866,
from the plates of 1840.

Kulik, 1848. The principal table occupies pp. 1-401, and gives the
squares and cubes of all numbers from 1 to 100,000. There is a compression
resembling that in Oekile's ' Eechentafeln ;' viz. the last four figures of the
square and cube are printed but once in each line, these figures being the
same for aU squares and cubes in the same line across the double page. The
arrangement will be rendered clear by the description of a page— say, that
corresponding to 92. There are ten columns headed ^2, 192, 292 992,

28 REPORT— 1873.

cack coHtaining two vertical rows of numbers, the one corresponding- to N'',
and the other to W ; the lines arc numbered 0, 1, 2. . . .49 (and on the next

double page 50 99). If, then, we wish to find the cube of 79217, we take

the figures 49711306131 from column 792, line 17, and add the last four
figures 1313 (which conclude the cube of 9217 in the same line); so that
the cube required is 497113061311313. Certain figures, common to the
■whole or part of a column, are printed at the top, and the change in the
column is denoted by an asterisk. This is the largest table of cubes in ex-
istence, and (except Lubole, which is of the same extent) is also the largest
table of squares. The printing is clear, and the book not bulky ; so that the
table can be readily used. At the end are eleven subsidiary tables. T. 1
(Perioden gerader Summenden) consists of columns marked 4, 6, 8 .... 48 at
the top, and 96, 94.... 52 at the bottom, each containing the "complete
period " of the number in question ; thus for 42 we have 42, 84, 26, 68, 10,
&c. (these numbers being the last two figures of a series of terms in arith-
metical progression, 42 being the common difference) ; and these are given
till the period is completed, i. e. till 42 occurs again. This may be at the end
of 25 or 50 additions ; if the former, the periods are given commencing
with 1 , 2, 3 (as well as 0) ; if the latter, with 1 or 2 only, as the case may
be; the periods for .r and 100 — x are of course the same, only in reverse
order. The use of the table as a means of verifying the table of squares
is obvious.

T. 2. Primes which are the sum of two squares (these being given also)
up to 10,529.

T. 3. Odd numbers which are the difference of two cubes (these being
given also) to 12,097.

T. 4. Odd numbers -which are the sum of two cubes (these being given also)
to 18,907.

T. 5-9. Four-figure additive and subtractive congruent endings for numbers
ending in 3 and 7, or 1 and 9, &e. : the more detailed description of these
tables belongs to the theory of numbers, -which wiU form a part of a subse-
quent Eeport.

T. 10. The 1044 four-figure endings for squares, and the figures in which
the corresponding numbers must end.

T. 11. First hundred multiples of tt and n~^ to twelve places. There is
appended to the tables a very full description of their object and use.

Bruno, Faa de, 1869. T. I. of this work (pp. 28) contains squares of
numbers from 0-000 to 12-000, at intervals of -001 to four places (stereo-
typed), intended for use in connexion -with the method of least squares.

The following are references to § 4 : —

Tables of Squares and Cubes, or both Squares and Cubes. — Schtoze, 1778
[T. IX.] and [T. X.] ; Hutton, 1781 [T. II.] and [T. III.] ; Vega, 1797,
Vol. II. T. IV. ; Lambert, 1798, T. XXXV. and XXXVI. ; Baklow, 1814,
T. I. ; Schmidt, 1821 [T. V.] (-with subsidiary tables) ; Hantschl, 1827,
T. VIII. ; *Salomon, 1827, T. I. ; Gbuson, 1832, T. II. and III. ; HtJLSsE's
Vega, 1840, T. IX. C. ; Teottek, 1841 [T. VI.] ; MUlleb, 1844 [T. III.] ;
iliNsiNGEE, 1845 [T. II.] ; KoHLEB, 1848, T. V. and VI. ; Willich, 1853,
T. XXI. ; Beabdmoee, 1862, T. 35 ; Eankine, 1866, T. I. and II. ;
Wackeebaeih, 1867, T. VI. ; Paekhuest, 1871, T. XXVI. and XXXII.,
and XXXIV. (multiples of squares); Petebs, 1871 [T. VI.]. See also
Tatloe, 1780 [T. IV.] (§ 3, art. 9).

Tables of Square Roots and Cube Roots. — Dodson, 1747, T. XIX. ;
ScHTJLZB, 1778 [T. XI.] and [T. XII.]; Maseees, 1795 (two tables);

ON MATHEMATICAL TABLES. 29

Vega, 1797, Vol. II. T. IV. ; Hanischi,, 1827, T. VIII. ; *Sai,omok, 1827,
T. I. ; Gruson.J 1832, T. IV. and V. ; HtrLsaE's Vega, 1840, T. VIII. ;
Trotiee, 1841 [T. VI.] ; Minsinger, 1845 [T. II.] ; Kohier, 1848, T. VII. ;
WiLLicH, 1853, T. XXI. ; Beaedmore, 1862, T. 35 ; *SonLOMiicH [1865?] ;
Rankinb, 1866, T. I. A ; Wackerbaeth, 1867, T. VII. See also Centnek-
scmvEE, 1825 [T. II.] (§ 3, art. 3). And for Squares (for method of least
squares), MtoLEE, 1844 [T. III.].

Endings of Squares. — (Three-figure endings) Lambert, 1798, T. IV.

Art. 5. Tables of Powers "higher than Cubes.
We know of no work containing powers of numbers (except squares and
cubes) only. Both Hfiton, 1781, and Barlow, 1814, give the first ten
powers of the first hundred numbers ; but we have seen no more extensive
table of this kind. Shanks (§ 4) gives every twelfth power of 2 as far as 2'^' ;
and, according to De Morgan, John Hill's 'Arithmetic,' 1745, has aU powers
of 2 up to 2"*. Tables of compound interest are, in fact, merely power tables,

as the amount of £K at the end of n years at r per cent, is M{ 1 + -^ ) . In

interest tables *• has usually values from 1 to 8 or 10 at intervals of | or |
for different periods of years ; but they could not be of much use, except for
the purpose for which they are calculated.

A good table of powers is stiU. a desideratum, as the need for it is often
felt in mathematical calculations. Very many functions are expansible in an
ascending (convergent) series of the form A^-\- A^oo -\- A.^x^ + &c., and a de-
scending series (generally semieonvergent) of the form 3^-{-'B^x;-^-\-'Bjic~^-\-
&c. The former is usually very convenient for calculation when oo is smaU,
and the latter wheh x is large ; but between the two, for values of x included
between certain limits above unity, there will be an interval where neither
series is suitable^the ascending series because the terms x, .%^,. . . . {x >1)
increase so fast that n must be taken very large (i. e. very many terms must
be included) before A„ is so small that A^x" can be neglected, and the de-
scending series because it begins to diverge before it has yielded as many
decimals as are required. Por these intermediate values the former series
(if there is no continued fraction available) must be used ; and then the terms
begin by increasing, often so rapidly, if x be moderately large, that it may be
necessary to calculate some of them to fifteen or twenty figures to obtain a
correct value for the function to only seven or eight decimals. In these
cases, so long as ten figures only are wanted, logarithms are employed ; but
when more are required recourse must be had to simple arithmetic ; and it is
then that a power table is so much needed. Mr. J. W. L. Glaisher has had
formed. in duplicate a table giving the first twelve powers of the first thousand
numbers, which, after the calculation has been made independently a third
time, will be stereotyped and published, probably in the coarse of 1873 ; it is
hoped that it will help to make the tabulation of mathematical functions
somewhat less- laborious and difficult.

The following tables on the subject of this article are described in § 4 : —
Tables of Poivers higher than Cubes. — Donsosr, 1747, T. XXT. (powers of 2)
and T. XXII.; Schueze, 1778 [T. VIIL] ; Hitiion, 1781 [T. IV.] ; Vega,
1797, Vol. II. T. II. (powers of 2, 3, and 5) ; Vega, 1797, Vol. II. T. IV. ;
L\MBEET, 1798, T. VII.-IX. (powers of 2, 3, and 5) and T. XL. ; Barlow,
1814, T. II. and III. ; Hulsse's Vega, 1840, T. VI. (powers of 2, 3, 5)
and T. IX. A, B, D, E ; Kohler, 1848, T. II. (powei-s of 2, 3, and 5) and
T. IV. ; Shanks, 1853 (powers of 2 to 2='^') j Beaedmoee, 1862, T. 35 j

30 REPORT— 1873.

Eankine, 1866, T. 2. See also Sir Jonas Moore [1650?], § 3, art. 4;
Taylor, 1781 [T. IV.] (§ 3, art. 9).

Tables for the solution of Cubic Equations, viz. +(« — a^). — LAMBiiiRT, 1798,
T. XXIX. ; Barlow, 1814, T. lY.

Art. 6. Tables for the expression of vulgar fractions as decimals.

The only separate tables we have seen are Wttcherer and Goodwtn's
works described at length below. The Babbage Catalogue contains the title
of an anonymous book, " Tafela zur Verwandlung aUer Briiohe von yi^ bis
^^W^> ^"^^^ ^°^ rum ^i* TiMMMTo ^^ fiinf- bis siebenziffrige Decimalbriiche,
4to, Oldenburg, 1842," of which De Morgan says "it gives every fraction
less than unity whose denominator does not exceed three figures, nor its nu-
merator two, to seven places of decimals. It is arranged by numerators ;
that is, aU fractions of one numerator are upon one double page." Eocipro-
cals would properly be included in this article ; but from their more frequent
use they have been placed in an article by themselves (§ 3, art. 7) ; Picahte's
table in that article gives multiples of reciprocals.

We must especially mention the " Tafel zur Verwandlung gemeiner
Briiche mit Nenncrn aus dem ersten Tausend in Decimalbriiche," which
occupies pp. 412-434 of vol. ii. of ' Carl Triedrich Gauss Werke,' Gottingen,
4to, 1863, and which somewhat resembles Goodwtn's tables described below.
In it, among other things, the reciprocal of every prime less than 1000 is
given completely {i. e. till the figures circulate). Had we met with the table
earlier we should have given a full description ; but we merely confine our-
selves here to giving the reference, reserving a more detailed explanation for
a future Eeport.

Wucherer, 1796. The decimal fractions (to five places) for all vulgar
fractions, whose numerators and denominators are both less than 60 and
prime to one another, arranged according to denominators; so that all
having the same denominator are given together ! thus the order is ... . -j'^,

T^T' If' • • ■ -tI' tV' "ft' ^^^ arguments being only given in their lowest

terms. After ^^ the system is changed, and the decimals are given for
vulgar fractions whoso numerators are less than 11 only; thus we have -^q,
■^' Tu- ■ • -io' sV' -^T- • • -^^ consecutive arguments (the arguments not being
necessarily in their lowest terms) ; and the denominators proceed from 50 to
999.

[T. II.]. Sexagesimal-Sriiche, viz. sexagesimal multiplication table to 60
X 60 ; thus, as 5 X 29" = 145" = 2' 25", the table gives 2.25 as the tabular
result for the joint-entry 5 and 29, There are seven other tables (II.-VIII.)
for the conversion of money into decimals of other money, for the coins of
diifereut countries ; the English table will serve as an example. There are

given as arguments ^, ^f „, -^ |ff (i. e. Id., 2d., &d., &c.), and as

tabular results the corresponding decimal fraction to ton places (i. e. of £1),
and also the shUlings and pence ; thus for -JJ- there are given •3041666666;
and Gs. Id.

Tho Beiehs-Oeld and Pfennig table is practically the same ; the denomi-
nators are in all cases 240, or 960, or submultiples of the latter. Regarded ma-
thematically the English table gives nearly as mnoh as all the rest, as for
denominators above 240 only a few numerators are talcen. There are also tables
of interest, present value, &c., to a great many places. The value of tt is given
on the last page to 306 places ; thus, if the diameter = 10000. . . .(306
piphers), then tt = 31415 (307 figures), the ciphers and figures being written

ON MATHEMATICAL TABLES. 31

at length — a cuiiona mode of statement at tlie end of a book occupied -with
decimal fractions.

' Goodwyn's Tables, 1816-1823. It is convenient to describe Good-
■wyn's four -works (tbe titles of which are given at length in § 5) together, as
they all relate to the same subject.

The Tabular Series of Deeimal Quotients (1823) forms a handsome table of
153 pages, and gives to eight places the decimal correapomding to every vulgar
fraction less than ^^, whose numerator and denominator are both not greater
than 1000. The arguments are not arranged according to their numerators or
denominators, but according to their magnitude, so that the tabular results
exhibit a steady increase from -001 (= Y(^^)to -09989909 (= JjiL). The
author intended the table to include all fractions whose numerators and deno-
minators were both less than 1000 without restriction ; and at the end of the
book is printed " End of Part I. ; " but no more was ever published.

The arrangement of the arguments in order of magnitude is not very good,
as it requires the first two figures of the decimal to be known in order to know
where to look for it in the table ; the table would be more useful if it were re-
quired to find a vulgar fraction (with not more than three figures in numerator
or denominator) nearly equal to a given decimal*; but this is not a trans-
formation that is often wanted. "When the decimal circulates and its period
is completed within the first eight figures, points are placed over the first and
last figures of the period, if not, of course only over the first; and by means
of the same author's table of ' Circles ' described below, the period can be
easily completed, and the whole deciinal fraction found. The fractions which
form the arguments are given in their lowest terms.

The TaUe of Circles (1823) gives all the periods of the circulating decimals
that can arise from the division of any integer by another integer less than
1024. Thus for 13 we find -676923 and -1 53846, which are the only periods
in which the fraction _ can circulate.

The periods for denominator 2" 5™ x are evidently the same as those for
denominator oo ; and arguments of this form are therefore omitted ; but a table is
given at the end (pp. 110 and 111), showing whether for any denominator less
than 1Q24 the decimal (1) terminates, and is therefore not included in the table,
(2) is in the table as it stands, or (3) is in the table but has to be sought
under a different argument (these last being numbers of the form 2" 5™ x).
A third table (p. 112) also gives the number of places after the separatrix
(decimal point) at which the period commences.

The principal table occupies 107 pp. Some of the numbers are very long,
(e. g., for 1021 there are 1020 figures in the period), and are printed in lines of
different lengths, giving a very odd appearance to many of the pagesf.

A table at the end contains all numbers of the form 2» 5™ that are less than

* It is proper to note, however, that the table was no doubt calculated for this purpose ;
the author considered his 'Table of Circles ' as giving decimals to vulgar fractions, and in-
tended this table to give vulgar fractions to decimals (see the introduction to the second
partof the 'Centenaiy' 1816); the ' Tabular Series ' (1816) is complementary to the 'Cen-
tenary;' but not so the 'Tabular Series' (1823) to the 'Table of Circles ' (1823), as the
latter only gives the periods.

t If the period of a decimal consists of an even number of figures, it is well known
tliat the figures in the last half are the comiplements to nine of the figures in the first
half; and the periods have been printed so that the complementary figures should be under
one another. When the period is odd, there is always another period of complementary
figures, and the two are printed one under the Other ; these facts account for what at first
sight appears a capricious arrangement of the figures. ,

32

REPORT — 1873.

1,000,000, arranged in order of magnitude, with the vahies of )i and m, and also
the values of the reciprocals of the numhers (expressed as decimals) and the
total numher of the proper vulgar fractions in their lowest terms which can
arise for any of the arguments as denominator. An examjjle of the use of
the tahles is given at the end of the hook.

The First Centenary S^e. [1816] contains the factors of all numhers to 100,
and the complete periods of their reciprocals or multiples of theii- reciprocals,
also the first six figures of every decimal fraction equivalent to a vulgar frac-
tion whose denominator is equal to the argument. The following is a spe-
cimen of one of the tahles :

34

2.17

•70588235
29411764

33

•970588

1

31

•9i.l764

3

29

•852941

5

27

•794117

7

25

•735294

9

23

•676470

11

21

•617647

13

19

■558823

15

The explanation is very simple: wo have If = •970588, and the other
figures of the period are 23529411764; ^\ = •9il764, and the other figures
are 70588235294, &c. If the numerator is in the third column wo take the
complement of the result (i. e. subtract each figure from 9) ; thus -Jj =
•029411, and the other figures of the period are 76470588235. The oven
numbers are omitted, as the fractions are not in their lowest terms ; thus ^^
= -j-f , and must be sought, under argument 17. [This table was published
separately by Goodwyn for private circulation. There is no date on the title-
page* ; but the address is written from Blackheath, and dated March 5, 1816.]
There is added a tabular series of complete decimal quotients of fractions
whose numerator is not greater than 50 and denominator not greater than
100 (the'heading of the table incorrectly says, "neither numerator nor de-
nominator greater than 100 "), arranged as in the ' Tabular Series' &c., 1823 ;
it is followed by an auxiliary table for completing such quotients as consist
of too many places to allow all the digits of their periods to appear in the
principal table. There is an appendix on Circulates &c. The ' Tabular Series'
(1816 and 1823) are interesting as exhibiting in the order of magnitude all
fractions whose numerators and denominators are both less than 100 up to j,
and whose numerators and denominators are both less than 1000 iip to ■^^^.
In the preface to the latter table the author gives as a fact he has observed, that

* It is by no means improbable that the titlepage has been torn out from the only copy
we have seen, viz, that in the Eoyal Society's Library.

ON MATHEMATICAL TABLES. 33

" In any three consecutive vulgar fractions in the table, if the numerators of
the extremes and the denominators be added together, the sum will form the
numerator and denominator of a fraction equal to the mean." That this is
the case with aU fractions, ranged in order, whose numerators and denomi-
nators are integers less than given integers, is a theorem discovered by Cauchy
and published by him in his ' Exercices.'

It has been thought worth while to describe Goodwyn's works at some
length, as they are almost unique of their kind, and are rarely to be met
with.

De Morgan states that " Mr. Goodwyn's manuscripts, an enormous mass
of similar calculations, came into the possession of Dr. Olinthus Gregory,
and were purchased by the Eoyal Society at the sale of his books in 1842."
There is no mention of them, however, in the Eoyal Society's Catalogue of
MSS. ; and nothing is known of them at the Society. They may possibly be
brought to light in the rearrangement of the manuscripts consequent upon the
approaching change of rooms.

Art. 7. Tables of JReciprocals.

The most extensive table is

Oakes, 1865. Eeciproeals from 1 to 100,000. This table gives seven figures
of the reciprocal, and is arranged as in tables of seven-figure logarithms ; viz.
the first four figures are found in the column at the left-hand side of the page,
the fifth figures run along the top line, and the sixth and seventh are inter-
polated for by proportional parts. The reciprocal of a number of five figures
is therefore taken out at once, and the process of taking out a reciprocal is
exactly similar to that of taking out a logarithm.

Prom 10,000 to 22,500 the differences and proportional parts (being
numerous) are placed on the lower half of the page, the differences being
also placed at the side of each line ; but above 22,500 the differences and
proportional parts are placed at the side of the page as in tables of logarithms.
The figures have heads and tails ; and the change in the third figure of the
reciprocal is made evident by prefixing an asterisk to the succeeding numbers
in the line. The table is the result of an original calculation, and was con-
structed by means of the obvious theorem that the difference of two recipro-
cals, divided by the difference of the corresponding numbers, is the reciprocal
of the product of those numbers. The reciprocals of the higher numbers,
however, were calculated by differences, which differences were found by
logarithms. Various checks were applied ; and the whole was virtually re-
computed on the Arithmometer of M. Thomas de Colmar. The significant
figures of the reciprocals alone are tabulated, decimal points and ciphers
being omitted, for the same reason that characteristics are left out in loga-
rithmic tables.

In T. I. of Babiow (§ 4) reciprocals are given of numbers from 1 to 10,000 ;
and this table also appears in the stereotype reprint of 1840 (see § 3, art. 4) :
the latter is the most generally used table of reciprocals, and is of _ sufficient
extent for most purposes ; it is also reputed to be very accurate, and is perhaps
free from error.

It must be added that Goodwyn's ' Table of Circles,' and ' Tabular Series,'
&c., 1823 (§ 3, art. 6), give reciprocals of numbers less than 1024 complete ;
viz. the whole period is given, even where it exceeds a thousand figures.

See also the reference to Gauss, vol. ii., near the beginniag of the last

article (§ 3, art. 6). . , ,. .

As most nearly connected with a table of reciprocals (it gives not only

1873. D

34 ampORT— 1873.

the reciprocals, but also multiples of them), wo here describe Pioaete's ' La
Division reduite h uiie Addition.'

Picarte[1861]. The principal table occupies pp. 15-104, and gives, to ten
significant figures, the reciprocals of all numbers from 1000 to 10,000, and also
the first nine multiples of the latter (which are therefore given to 10 or 11 sig-
nificant figures) . It is easy to see how this table reduces Division to Addition .
The arguments run down the left-hand column of the page ; and there are nine
other columns for the multiples ; each page contains 100 lines ; so that there
are 10,300 figures to the page. Owing, however, to its size, and to the smallness
and clearness of ihe figures, there is no confusion, the lines being well leaded.
The great table is preceded by two smaller ones, the first of which (pp. 6, 7)
gives the figures from the ninth to the fourteenth (inclusive) of the logarithms
of the numbers from 101,000 to 100,409 at intervals of unity (downwards),
with first, second, and third differences ; and the second (pp. 10, 11)' gives
ten-figure logarithms of numbers to 1000 ; and from 100,000 to 101,000 at in-
tervals of unity (with differences). There is also some explanation &c.
about the manner of calculating . logaiithms by interpolation, &o. The
author remarks on the increasing rarity of ten-figure tables of logarithms,
referring, of course, to Vlacq and Veoa. The whole work was submitted by
its author to the French Academy, and reported on favourably by a Commit-
tee consisting of MM. Mathieu, Hermite, and Bienayme. The report (made
to the Academy Feb. 14, 1859) is printed at the beginning of the work.
M. Eamon Picarte describes himself as Member of the University of Chili ;
and the Chilian Government subscribed for 300 copies of the work. There
is no date ; but the " privilege " is dated Nov. 1860, and the book was re-
ceived at the British Museum, April 29, 1861, so that the date we have
assigned is no doubt correct. On the cover of the book are advertised the
following tables by the same author, which we have not seen : — •

" Tables de multiplication, contenant les produits par 1, 2, 3 . . . . 9 et toutes
les quantites au-dessous de iO,000, 1 vol. in-18 j&us."

" Tableau Pithagorique, etendu jusqu'tl 100 par 100, sous une nouvelle
forme qui a permis de supprimer la moitie des produits.".

It is scarcely necessary to remark that any trigonometrical table giving
sines and cosecants, cosines and secants, or tangents and cotangents, may be
used (and sometimes with advantage) as a table of reciprocals. The extreme
facility with which reciprocals can be found by logarithms has prevented tables
of the former from being used or appreciated as much as they deserve.

The following is the list of references to § 4 : —

Tables of Reciprocals.— 'Kk^^w.s, 1795; Barlow, 1814, T. I. (to 10,000) ;
Thotteb, 1841 [T. VIII.] ; Wmicu, 1853, T. XXI. ; Beabdmoee, 1862, T.
35 ; ScHLOMiiCH [1865 ?] ; Eaneine, 1866, T. I. and I. A ; Wackereaeth,
1867, T. IX. ; Parkhtjest, 1871, T. XXV. ; see also Mekpaut, 1832 (§ 3^
art. 3),- Baemw (1840) (§ 3, art. 4).

Art. 8. Tables of Divisors (Factor tables), and Tables of Primes.

If a number is given, and it is required to determine whether it be prime,
and if not what are its factors, there is no other way of effecting this ex-
cept by the simple and laborious process of dividing it by every prime less
than its square root, or until one is found that divides it without remainder*.
The construction of a table of divisors is on the other hand very simple, as it

* Wilson's theorem (viz. that 1 . 2 . 3 ....(» — 1) + 1 is or is not divisible by n,
according as n.ia or is not prime) theoretically affords a criterion; but tlie labour of
applying it would be far greater than the direct procedure by trinl, ,

ON MATHEMATICAL TABLES. 35

is merely necessary te form the multiples of 2, 3, 5 . . up to tte extent of the
table, the numbers that do not occur being of course primes. The manner
in -which the formation of these multiples is best effected, and other practi-
cal details, are explained by Bttkokhakdi in his preface to the second
million. The following is a list of tables of divisors and of primes, abridged
from an elaborate account prefixed to Cheenao : —

1657. Francis Sohooten : table of primes to 9997.

1668. Pell (in Branker's translation of Ehonius's ' Algebra,' published at
London) : least divisors of odd numbers not ending in 5 to 100,000.

1728. Poetius. An ' anatome.' of numbers to 10,000.

1746. KRiJsEK. Primes to 100,999.

1767. Anjema. All divisors (simple and compound) of numbers to
10,000.

1770. Lambekt. Least divisors of numbers to 102,000 (multiples of 2, 3,
and 5 omitted).

1772. Marci. Extension of Lambert's table by the addition of primes to
400,000.

1785. Neumann. Simple divisors (Pell only gave the least) of numbers
to 100,100 (multiples of 2, 3, 5 omitted).

1797. Vesa. Simple factors to 102,000, and primes to 400,000' (see
Tega, ' Tabulse,' 1797, Vol. II. T. I.). ' '

1804. Krause. Factor table to 100,000.

From the above list Chernac has omitted Rahn (1659), giving factors to
24,000, and Pigki (1758) to 10,000, which are described below. A more
important omission is that of Felkel, whose table is noticed at lengt!h
further on.

The titles of Anjema's, Neumann's, and Krause's works are given in the
Babbage Catalogue as follows : — " Anjema (Henricus), Tabula divisorum
omnium numerorum naturalium ab 1 usque ad 10000. 4to, Lugd. Bat.
1767 ; " " Neumann (Johann), Tabellen der Prim-Zahlen und der Factoren
der Zahlen, welche unter 100100, und durch 2, 3, oder 5 nicht theilbar sind ;
herausgegeben durch J. N. 4to, Dessau, 1785;" and "Krause (Karl C. F.),
Factoren- und Primzahlen-Tafel von 1 bis 100000 neu berechnet. Fol.
Leipzig, 1804,"

The same catalogue also contains the title, " Snell (F. "W. D.), Ueber eine
neue und bequeme Art, die Faktorentafeln einzurichten, nebst einer KUp-
fertafel der einfachen Faktoren von 1 bis 30000. 4to. Giessen und Darm-
stadt, 1800."

The following are accounts of tables we have seen ; —

Rahn, 1659. On pp. 37-48 is given a table of divisors; viz. the least
divisor of every number, not divisible by 2 or 5, is tabulated from 1 to 24,000,
the primes being marked with a p.

Pigri, 1758. All the simple factors (so that if multiplied together they
give the number) are given of aU numbers from 1 to 10,000. "When the
number is a power, letters are used instead of numbers (a = 2, 6 = 3, c = 5,
&c., as explained on p. 11 of the book) ; thus, answering to 25 we have ec,
to 27 lib, to 225 66, cc, &c.

Kriiger, 1746. At the end of the ' Algebra ' is a list of primes to 100,999,
arranged consecutively in pages of six columns, and occupying-47 pp. The
titlepage runs ' Primzahlen von 1 bis 1000000' ; but the limit is as above
stated ; a;nd there is no possibility that the copy before us is incomplete, as the
last page is a short one, and there is no printing on the back.

D 2

36 iiEi'oiiT— 1873.

The primes of each hundred are separated, -which for some purposes would
be an advantage.

Lambert states (Introd. ad ' Supplemental (fee, 1798) that KEtGBK received
this table from Peter Jseger.

Felkel, 1776. Table of all simple factors of numbers to 144,000, the
tabular results being obtained from three tables. Thus Table A gives primes
to 20,353 ; these occupy one page, along the top line of which run the Greek
letters a, /3. .'. . and down the left-hand column four alphabets consecutively,
viz. small italic, small German, capital italic, and capital German (there
being 100 lines); and any prime given on this page is henceforth in the book
denoted by its coordinates, so to speak : thus 9839 would be printed fip, &c.
The principal table occupies 24 pp. ; and then Table B occupies one page at
the end. Suppose it required to find the factors of 138,593. The . middle
table is entered at 138 and Table B at 593. In the latter we find as result
"^r line 20," so that we know that the compartment under </ in the 20th Hue of
the block 138, refers to the number in question. In this compartment is printed
«, g, /3r, which, interpreted by Table A, gives 7, 13, and 1523 as the factors.
There are a few details that have been omitted in this description ; the last
three figures are written in the compartment wherever there is room for
them.

On the titlepage is a large engraving of a student (no doubt a portrait of
Pelkel) turning in contempt from a disordered cabinet of military books to
another neatly arranged, containing Euler, Newton, Madaurin, Bernoulli,
Bosoovich, &c., and holding in his hand the works of Lambert ; with mottoes
" Bella odii Pacem diligo, vera seqnor," &c. above. It will be seen that this
table is entirely superseded by Chernac and Burckhardt. In the arrangement
of the latter the table would only have occupied 16 niuCh smaller pages,
and its use would have required no explanation ; but on account of the rarity
of the work, it has been thought worth while to describe at some length
what is certainly the most remarkable-looking table we have seen.

De Morgan states that " Murhard mentions the first part of a table (by
A. Felkel) of the factors of all numbers not divisible by 2, 3 or 5 from 1 to a
hundred millions, Vienna (1776)." On referring to Murhard we find such is
the case, " 100,000,000 " being an obvious misprint for " 10,000,000 ; " we
have seen Murhard's error reproduced by other writers.

Of Felkel's table Gauss (in the letter prefixed to Base's Seventh Million)
Bays : " Felkel hatte die Tafel im Manuscripte bis 2 Millionen fertig und dor
Druck war bis 408,000 fortgeschritten, dann abcr sistirt, und die ganze
Auflage wurde verniohtet bis auf wenige Exemplare des bis 336,000 gehenden
Theils, wovon die hiesige Bibliothek eines besitzt." The copy of Eelkel in
the Eoyal Society's Library, which extends to 144,000, is that which has
been described above. Pelkel's table is also referred to by Hobeei and
Idblee in the introduction to their work (see § 4).

Eelkel was editor of the Latin edition (Lisbon, 1798) of Lambert's
'Zusatze' (the ' Supplcmenta' &c., see § 4) ; and he has, there given, in the
• Introductio Interprotis' and at the end, some account of his life and the work
he accomphshed and hoped to accomplish with regard to the theory of numbers.
Ho commenced the study of mathematics when of a somewhat advanced age •
and he speaks in the warmest terms of Lambert, with whom ho was in cor-
respondence, -and from whom he derived much assistance. This accounts for
Lambert being the book open before the student in the engraving described
above.

In a note on p. xir of the Introductio to the • Supplemental he (Eelkel)

ON AlATHEAIATICAL TABLES. 37

Says : " Non solum inveni formam omnes divisores immerorum excopto maxi-
mo, ah 1 usque 1,008,000 in spatio 42 plagularum reprsesentandi, verum etiam
rcipsa opus spatio 16 mensium usque ad 2,016,000 confeci, aunoque 1785
.... ad 5,000,000 usque continuavi." (See also p. vii of the ' Introductio In-
terpretis ').

Since writing the above description of Felkel, I have examined (in the
Graves Library) a far more complete copy, -which contains probably all that
Felkel over printed. There are three parts (bound together). The first is the
same as that described above, and extends to 144,000 ; the second part
(with fresh pagination) extends from 144,001 to 336,000 (pp. 2-63) ; we
then have 'Tabula Factorum pars III exhitens factores numerorum ab
336,001 usque 408,000/ occupying pp. 66-87. The table thus gives factors
as far as 408,000. The words « 336,001 usque 408,000 " have clearly ori-
ginally stood " 144,001 usque 366,000 ;" but the latter numbers have been
stamped out and the former printed over them. There is a note in the work
in the handwriting of Mr. Graves's librarian, ■ which, referring to Gauss's
remark quoted above, proceeds : — " This copy contains 3 parts and gives the
factors of all numbers up to 408^000; such a copy is perhaps unique."
Gauss stated that all the copies were destroyed except a few, which extended
to 336,000 ; so that there can be no doubt that the Graves copy, extending
to 408,000, must be, to say the least, excessively rare.

It should be added that the title and preface to the Graves copy are in
Latin, while the Koyal Society's copy has them in German (Poggendorff
also quotes the title in German with date 1777) ; the preface is dated April 1,
1777, although the titlepage bears date 1776. In the Graves copy some
errata in Part I. are given.

For several reasons Felkel's connexion with numerical tables is a curious
one, and the record of his life would be interesting. We have seen (in some
work of reference) a number of mechanical contrivances assigned to him as
their inventor.

Chernac, 1811. . In a thick quarto are given all the simple divisors of
numbers from.l to 1,020,000 (multiples of 2, 3, and 5 being excluded).
This book was found by Burckhardt (who subsequently published the same
table, the least divisor only being given) to be very accurate ; he detected only
38 errors (he has given them in the preface to his first million), of which only
9 are due to' the author, the remaining 29 having been caused by the slipping
&c. of type in the printing. . , ■

Hutton's Phil, and Math. Diet. 1815. In vol. ii. pp. 236-238 (Art.
' Prime Numbers ') is a table giving the least divisor of all numbers from 1 to
10,000, multiples of 2 and 5 being omitted.

Burckhardt (First Million), 1817. Least divisors of every number to
1,020,000. The library of the Institute contained a manuscript (calculated
by Schenmarck ?) giving the least divisor of numbers to 1,008,000 ; Burck-
hardt therefore computed the next 12,000 himself, and compared the manu-
script with Cheknao — a laborious work, as when a wrong divisor was given,
Burckhardt had to satisfy himself if the number was really prime, as was
the case in 236 instances. For primes less than 400,000 he referred to Vega
(see Vega's 'Tabula,' 1797, Vol. II. T. I., and Htri.ssE's Vega, 1840, T. V.).
Only 38 errors were found in Cheenac. On the last page is a smaU table con-
taining the number of figures in the periods of the reciprocals of 794 primes
below 9901 (779 of which are below 3000). Burckhardt mentions in the preface
-that he has nearly completed the manuscript of the fourth, fifth, and sixth
millions, which will be published, if the sale .of the first three millions is

38 REPORT — 1873.

sufficiently favourable to induce the bookseller to undertake them. There
arc three pages on the use of the tables. This work, though containing the
first million, was published after the second and third.

Five errors are pointed out at the beginning of Base's/ Seventh Million.'

Burckhardt (Second Million), 1814. ■ The arrangement is the same as for
the first million ; and the table extends from 1,020,000 to 2,028,000. This
was the first published of the three millions ; and the method of calculation &c.
is explained in the introduction, the least factor alone being given. If the
others are required, the process is of course to divide the number by this factor
and enter the table again with the quotient. To facilitate the division, on
the first page (p. -viii) a table is given of the first 9 multiples of all primes
to 1423.

Burckhardt (Third Million), 1816. The arrangement is the same as in
the other millions : the table extends from 2,028,000 to 3,036,000.

Rees's CyclopBedia(vol..xxviii. Art. 'Prime Numbers'), 1819. Attached
to the article "Prime Numbers" in Kees's ' Cyelopsedia,' is a table of 23 pp.,
giving a list of primes up to 217,219 arranged in decades — a very convenient
table, as there are 910 primes on each page. It is stated (and truly) that the
primes are given to twice the extent that they are to be found in any previous
English work. In the course of the article the author says, "And a work lately
published in Holland, not only contains the prime numbers up to 1,000,000,
but also the factors of all composite numbers to the same extent— a performance
which, it must be allowed, displays the industry of its author to much more
advantage than either his genius or judgement." This can only refer to Chee-
KAc's table, which was published at Deventer (Daventria) in 1811 ; and it is a
matter of regret that an English writer on mathematics should have thought
only deserving of a sneer a work the performance and extension of which
had been consistently urged by Euler and Lambert and afterwards by Gauss.
One would expect the article of such a writer on the theory of numbers to be
very poor ; and such is the case. He has not thought it worth while to
state where the table he gives has been copied from ; it is no doubt taken
from Vega (' Tabula '), 1797, Vol. II. T. I.

Dase (Seventh Million), 1862. The least divisor of all numbers from
6,000,001 to 7,002,000 (multiples of 2, 3 and 5 excluded), and therefore
also a table of primes between these limits.

The arrangement is as in Btjkckhaedt, there being 9000 numbers to the
page.

This work was undertaken by Dase at the suggestion of Gauss ; and the letter
of the latter is printed in the preface. In it Gauss adverts to, and expresses
his concurrence in, Felkel's desire ihat the factorial tables should be extended
to ten mUHons ; he states that a manuscript containing the fourth, fifth, and
sixth millions (viz. 3,000,000 to 6,000,000) was some -years before presented
hy CreUe to the Berlin Academy, and he expresses a hope that it will soon be
published ; he therefore suggests that Base should complete the portion
from 6,000,000 to 10,000,000. Dase accordingly undertook the work, and
at the time of his death in 1862 had finished the seventh mOHon entirely
and the eighth million nearly j while many factors for the ninth and tenth
.millions had been determined. The seventh million (as also the two follow-
ing) were published after Dase's death by a committee of his fellow-towns-
men as a memorial of his talent for calculation.

Dase (Eighth Million), 1863. The arrangement is the same as in the
seventh million ; and the table extends from 7,002,001 to 8,010,000 ; the
.paging runs from 113 to 224.

ON MATHEMATICAL TABLES. 39

. There is a short preface of 2 pp. by Dr. Eosenberg, who edited the work,
which was left nearly complete by Dase.

Dase and Rosenberg (Ninth Million), 1865. The arrangement is the
"same as in the previous two millions ; and the table extends from 8,010,000
to 9,000,000. The work left incomplete by Dase at his death was finished
by Dr. Rosenberg ; the paging runs from 225-334.

It is stated in the preface that the tenth million (the last which the tables
were intended to include) was nearly completed ; but we believe it has not
yet appeared.

It will have been seen from the above accounts that Chbenao's, Btook-
HABDi's, and Dasb's tables together contain all the published results with re-
gard to factors of numbers ; and by means of them we can find aU. the
simple divisors of numbers between one million and three millions and
between six millions and nine millions easily, and between unity and one
million at sight. There is, however, the gap from three millions to six
millions; and it is very much to be regretted that this is not filled up.
'Gauss states a table of divisors from three millions to six millions exists in
manuscript at Berlin ; and Burokhardt also formed a similar table ; so that
.this portion has apparently been twice calculated (by. Crelle? and Burok-
hardt). . , , .

Gauss's. letter is dated 1850 ; and it is a calamity that the anticipations con-
tained in it have not been realized, as a manuscript unpublished does more
'harm than if it were non-existent, by checking others from attempting the
.task. The completion of Gauss's scheme (viz. the publication of tables to ten
millions) is very desirable, as these tables may be regarded as data in regard
ta investigations in the theory of numbers (see references to memoirs of Euler
and Gauss in Chebnac, and Gauss's letter). The tenth miUion also seems to
,be still unpublished, though seven years ago we had Dr. Eosenberg's assurance
that it was nearly completed. If the whole ten millions were published, we
should much like to see a list of all the primes up to this point published
separately.

Oakes, 1865 (Machine table). The object is to find the prime or least
factors of numbers less than 100,000 j and for this purpose there are three
itables, A (1 page large 8yo), B (4 pp. folio), and (1 page obi. foHo), and
nine perforated cards, the one to be em.ployed depending on the group of
,10,000 th3,t contains the argument. The mode of entry is somewhat compli-
cated ; and the table can only be regarded as a matter of curiosity ; for in the
method of arrangement of Btjeckhakdt or Dase the least factors of all
numbers under 100,000 only occupy a little over 11 pp. or six leaves
.of smaU folio or large 8vo size — while the present apparatus consists of six
leaves of large and difierent sizes, and nine cards, besides requiring an
aiiyolvfed course of ptocedure. Col. Oakes does not explain the principle
on vehich his method depends. _ ,

" The fQll'Qwiiig is a list of tables contained in works that are described in

S 4 , ; . . .

Tables of Divisors:— DomOi^; 1747, T. XVII. (to 10,000) ; Masebbs, 179^
(to 100,000) ; Vecja, 1797, Vol. II. T; I. (to 102,000) ; Lambebt, 1798,
T. I. (to 102,00Q); Bablow, 1814, T. I. (to 10,000); Hamschi, 1827,
T VII (to 18,277) i *Salomon, 1827, T. 11. (to 102,011); HUlsse's Vesa,
1840, T. V, ; K6hi,eb, 1848, T, VIII. (to 21,524) ; Houul, 1858, T. VII. (to
10,841) ; Eaitktne, 1866 (to 25§). 'See also Gbitson, 1798, § 3, art. 1. ,

List of Prime i\%m6«r«.— Dobson, 1747, T. XVIII. (10,000 to 15,000);
VESA, 1797, Vol. II. T. I. (102,000 to 400,000); Lambebt, 1798, T. II.

40 REPOILT— 1878.

(multiples of primes); T. VI. (to 102,000); Bahtow, 1814, T. V. (to
100,103) ; Ht'LssE's Yega, 1840, T. V. (102,000 to 400,313); Minsingek,
1845 [T. II.] (to 1000) ; ByEiTE, 1849 [T. I.] (to 6000) ; WACiasEBAKin,
1867 (to 1063) ; PABKnuEST, 1871, T. XXIII. (to 12,239).

Art. 9. Sexagesimal and Sexcentenary Tables.
Originally all calculations were sexagesimal; and the relics of the S5'stem
still exist in the division of the degree into 60 minutes, and the minute into
60 seconds. To facilitate interpolation, therefore, in trigonometrical and
other tables, several large sexagesimal tables have been constructed, which
are described or referred to below. They are, we believe, scarcely used at
all now, for several reasons — first, on account of the somewhat cumbrous size
of the complete tables, and secondly because for most purposes logistic
logarithms (see § 3, art. 18) are found more expeditious and convenient. A
third reason is Uiat both Bbrnoitlli's and Taylor's tables were published by
the Commissioners of Longitude, and, like the other publications of the Board,
were advertised so little that their existence never became generally known.

Bernoulli, 1779. A sexcentenary table to 600 seconds, to evei-y second,
giving at once the fourth term of any proportion of which the first term is
■600" and each of the other two are less than 600", The table is, of course, of

double entry ; it may perhaps be best described as giving the value of ^Jir;,

correct to tenths of a second, x and y each containing numbers of seconds
less than 600", .r being expressed in seconds alone, and y in minutes and
seconds (though the latter can be turned into seconds at sight, as the number
of seconds in the necessary integer number of minutes is given at the top of
each page). The x's run down the left-hand column, and the j/'s along the top
line ; and the arrangement is thus : — The portion of a? from 1" to 60" and the
whole range of y is given ; this occupies 30 pp. ; then the portion for x from
60" to 120", and for y from 60" to 600"; and so. on. The chief use of the
table consists in the fact that in astronomical tables the differences are
usually given for every 10', so that the interpolation gives rise to a proportion
of the kind described above : in some cases the use of the table would be
preferable to that of logistic logarithms.

Taylor, 1780 [T. I.] (pp. 240). The table exhibits at sight the fourth
term of any proportion where the first term is 60 minutes, the second any
number of minutes less than 60, and the third any number of minutes and
seconds under 60 minutes. If the second term consists of mimites and seconds,
the table must be entered twice (once for the minutes and once for the seconds).
The table can of course also be put to other uses.

- There is also added a table of the equation of second difference, giving the
correction to be applied on this account in certain cases.

[T. II.] (pp. 250, 251). Giving the thirds answering to the decimals in
every column of [T. I.] where the result is expressed in minutes, seconds, and
decimals of a second.

[T. III.] (pp. 263-312). A millesimal table of proportional parts adapted to
sexagesimal proportions, giving the result of any proportion in which the first
term is 60 minutes,-the second term any number under 60 minutes, and the
third term any absolute number under 1000. It is in fact the same as the
sexagesimal table [T. I.], only that the third term is expressed in seconds,
and is given only to 1000 (16' 40"), and the result is also expressed in
seconds (in [T. I.] the third terms are given both in minutes and seconds) and

ON MATHEMATICAL TABLES. 41

in seconds wholly, so that the expression of the result in seconds wholly is the
chief characteristic of [T. III.].

This table is followed by 3 pp. to convert sexagesimals into decimals and
vice versa, and numbers iuto sexagesimals and vice versd. The other tables
arc weights and measures &c. There are numerous examples given in the
introduction.

[T. IV.]. Another table occupying one page (p. 252) should be noticed ;
it gives squares, cubes, fourth, fifth, and sixth powers of any number of
minutes up to 60' : thus the square of 3' is 9" ; the cube, 27'" ; the fourth
power 1'" 21" ; the fifth 4'' 3', &c. The words sursolid and sqimre cube are
xised for the fifth and sixth powers.

On the present work see also Bbveelet (1833?) (§ 4).

It was the author of this table (Taylor) who afterwards calculated the
logarithmic trigonometrical canon to every second.

The following are references to works in § 4 : —

Sexacjesimca tables :— Ltnw, 1827, T. ^ ; Babat, 1829, T. XXIV. (lo-
garithms with sexagesimal arguments); TJiiVEBLET(1833?),T. VI. (pp. 232
&c.) and T. XV. ; Shoeteede (Com. log. Tab.), 1844; Goedon, 1849, T.
XVII. (half sines, &c., expressed sexagesimally).

Tables for the conversion of sexagesimals into decimals, and vice versd: —
■DoTOiAS, 1809, T. III., Supplement ; Ditcom, 1820, T. XX. ; Htjlsse's
Vega, 1840, T. IV.

Art, 10. Tables of natural Trigonometrical Functions.

A history of trigonometrical tables by Hutton is prefixed to all the editions
of his ' Tables of Logarithms ' published during his lifetime * ; and, in his
Article on Tables in the ' English Cyclopaedia,' De Morgan has given what
is by far the most complete and accurate account of printed tables of this
kind that has been published. Information about the earlier tables is also
to be fonnd in Montucla and Delambre (see references in De Morgan). For
many years, when Mathematics had not passed beyond Trigonometry,
the method of construction and calculation of the ' Canon Trigonometricus *
formed one of the chief objects of the science, and the works on the subject
were comparatively numerous, though now, of course, of purely historic
interest only. Prior to the introduction of siues from the Arabians by
Albategnius, trigonometrical calculations were always made by chords. The
unit-arc was the arc whose chord was equal to the radius (viz. 60°) ; and
both arc and radius were divided into 60 equal parts, and these subdivided
again into 60 parts, and so on. (It thus appears that it was not the right
angle that was divided into 90, 60 and 60 parts, &c., but that the unit-angle
was 60°, so that the division was strictly sexagesimal throughout. It is
curious that in some modern tables (see Beveeley, T. VI. and XV. &c.) the
original arrangement has been restored, for convenience of interpolating by
TAyroK's sexagesimal table). Thus in the earliest existing table, viz. the
table of chords in the Syntaxis of Ptolemy (died a.d. 178), the chord of 90°
is 84° 51' 10". Purbach (bom 1423) and Eegiomontanus (born 1436) calcu-
lated sines, the former to radius 600,000 and the latter to the same radius
and also to radius 1,000,000; but it is not certain whether they were printed.
The first known printed table, according to De Morgan, is a table of sines
to minutes, without date, but previous to 1500. Peter Apian first published
a, table with the radius divided decimally (1533). Tangents were first pub-

* It also forms Tract XIX. vol. i. pp. 278-306 of his ' Mathematical Tracts,' 1812.

43 REPORT— 1873.

lished by Begiomontanus (1504) ; and the first complete canon giving all the
six ratios of the sides of a right-angled triangle is duo to Ehetions (1551),
who also introduced the semiquadrantal arrangement. Eheticus's canon was
to every ten minutes to 7 places ; and Vieta first extended it to every minute
(1679). The first complete canon published in England was by Blundevile
(1594), although a table of sines had appeared four years earlier.

It may be added that Eegiomontanus (1504) called his table of tangents (or
rather cotangents) Tabula fcscwnda, on account of its great use ; and till the in-
troduction of the word tangent by Finck (1583), a table of tangents was called
a Tabula foecunda or Canon fmcundus ; Finck also introduced the term secant,
the table of secants having previously been called Tabula henefica by Mauro-
lycus (1558), and Tabula fcecundissima by Vieta.

■ The above historical sketch has been compiled from Hutton and De Morgan ;
so that most of the statements contained in it are not derived from our own
inspection of the works mentioned. It is only inteuded to give an idea of the
history of the natural canon ; and from the experience we have had of the value
of second-hand information in mathematical bibliography, we should not re-
commend great reliance to be placed on any one of the facts. A good deal of
information about Eheticus, Vieta, &o. is given by De Morgan, whom we have
sca:rcely ever found inaccurate, even in trifling details, when describing works
he has examined himself. "We have seen several of the works noted, but not
Bufiicient to make any corrections of importance to the current histories.

The next author of importance to Eheticus was Pitisctjs (1613), whose im-
portant canon, which still remains unsuperseded, is described below. The in-
vention of logarithms in the following year changed all the methods of calcula-
tion ; and it is worthy of note that Napieb's original table of 1614 (see § 3, art.
17) was a logarithmic canon of sines and not a table of the logarithms of
numbers. Almost at once the logarithmic superseded the natural canon ;
and since Pitisctjs's time no really extensive table of pure trigonometrical
functions has appeared. Natural canons are now most common in Nautical
collections, where the tabular results are generally given to 5 or 6 places only.
, Traverse tables (multiples of sines and cosines) have not been included
(see § 2, art. 12). Massaloup (described below), however, is really a table
of this kind, although constructed for a different purpose.

Finck [1583]. Canon of sines, tangents, and secants in separate tables,
quadrantaliy arranged, for every minute of the quadrant, to 7 decimal places.
The sines occupy pp. 138-173, the tangents . pp. 176-221, and the secants
pp. 224-269. De Morgan says that Finck calculated his own secants. There
is no date on the titlepage ; but the preface and the colophon are both dated
1583. The name tangent is introduced by Mnck on p. 73, and that of
secant on p. 76. These names were speedily adopted : thus Clavius, at the
.end of his edition of ' Theodosius ' (Eome, 1586), reprints Finck's tables, and
nses his terms both in the headings of the tables and in the trigonometry.
He does not mention either Finck or Eheticus by name, but speaks of them
as reeentiores (p. 188). Pitiscus, in his trigonometry appended to Abraham
Shultet's ' Sphaericorum ' (Heidelberg, 1595), uses the names tangent and
secant, and refers to Finck or Eheticus for the requisite canons ; and in his
larger trigonometry (Augsburg, 1600) he reprints Finck's tables to five deci-
mals, placing the sines, tangents, and secants together in one table. Blun-
devile, in his ' Exercises ' (London, 1594), reprinted the tables from Clavius.
All these works are before us ; but a more detailed account would be of only
historical or bibliographical interest.

ON MA.THEMATICAL TABLES. 43

- Hheticus, 1596^ (' Opus Palatinum '). Complete ten-decimal trigonome-
trical canon for every ten seconds of the quadrant, semiquadrantally arranged,
with dififerences for all the tabular results throughout. Sines, cosines, and
secants are given on the versos of the pages in columns headed respectively
Perpendiculum , Basis, Hypotenusa ; and on the rectos appear tangents, cose-
cants, and cotangents, in columns headed respectively Perpendiculum, Hypo^
tenusa, Basis*. This is the celebrated canon of George Joachim Eheticus,
the greatest of the table-computers, to whom also is due the canon of sines
described below under Piiiscus, 1613. At the time of his death (1576)
Eheticus left the canon all but complete ; and the trigonometry was finished
and the whole edited by Valentine Otho under the title ' Opus Palatinum,'
so-called in honour of the Elector Palatine Frederick IV., who bore the ex^
pense of publication. The edition before us is in two volumes, the second
containing the ten-decimal canon and occupying 540 pp. (2-541) folio ; then
follow 13 pp. of errata numbered 142-153 and 554. At the end of the
first volume is a canon of cosecants and cotangents (in columns headed
Hypotenusa and Basis respectively) to 7 places for every 10 seconds, in a
semiquadrantal arrangement. It occupies 180 pp. (separate pagination,
2-181) ; and there seems no reason why it should have been printed at all, as
the great ten-decimal canon completely supersedes it. Besides, it is exceed-
ingly incorrect, as comparison. with the latter shows at once. On this point
Do Morgan says that its insertion " was merely the editor's want of judg-
ment ; it is clearly nothing but a previous attempt made before the larger
plan was resolved on ;" while Hutton writes, " But I cannot discover the
reason for adding; .this, less table, even if it were correct, which is far from
being the case, the numbers being uniformly erroneous and different from the
former through the greatest part of the table." Mention of it is introduced
by Hutton with the words, " After the large canon is printed another smaller
table," Ac, while in the copy before us it ends the first volume, the second
containing the great canon. It is also to be inferred from De Morgan's ac-
count that the whole work generally is bound in one (very thick) volume.
The tangents and secants in the early part of the great canon were found to
be inaccurate ; and the emendation of them was intrusted to Pitiscus, who
" corrected the first eighty-six pages, in which the tangents and secants were
sensibly erroneous " (De Morgan) j and copies of this corrected portion alone
were issued separately in 1607, as well as of the whole table with the correc-
tions. We have not seen one of these corrected copies ; but vide De Morgan's
full account, ' English Cyclopaedia,' Article " Tables," and ' Notices of the
Eoy. Astron. Soc.,' t. vi. p. 213, and ' Phil. Mag.' June, 1845. The pagina-
tion of the other parts of the work is ' De Triangulis globi cum angulo recto,'
pp. 3-140 ; ' De Fabrica Canonis,' pp. 3-85 ; ' De Triquetris rectarum line-
arum i;i planitie,' pp. 86-104 ; ' De Triangulis globi sine angulo recto,' pp.
1-341 ; ' Meteoroseopium,' pp. 3-121 (the first three by Eheticus and the
rest by Otho).

In 1551 Eheticus had published a ten-minute seven-place canon in his
' Canon Doctrinae Triangulorum,' Leipzig, with which the present work must
not be confounded. And in 1579 Vieta published his ' Canon Mathematicus,
seu ad triangula cum Adpendicibus,' for every minute of the quadrant. This

* The explanation of these terms is evident. The sines and cosines are perpendiculars
and bases to a hypotenuse 10,000,000,000; the secants and tangents are hypotenuses
and perpendiculars to a base 10,000,000,000, and the cosecants and cotangents are hypo-
tenuses and bases to a perpendicular 10,000,000,000. The object Eheticus had in view
■jras. to calculate the ratios of each pair of the sides of a right-angled triangle..

44 REPORT — 1873.

and several other works that wc have examined will be noticed at length in a
future Eeport.

On Rheticus's other works see Piiiscirs, 1613, below.

Gernerth has given a list of 598 errors that he found in the first seven or
eight figures of the ten-decimal canon in the ' Zeitsehi-ift f. d. osterr. Gymn.'
VI. Heft, S. 407 (also published separately). He also gives an account of the
contents of the ' Opus Palatinum,' from -which it appears that in his copy the
different parts of it were bound up in a different order from that in which they
appear in the copy we have examined (which seems to bo anomalous in this
respect) ; and he omits the 121 pp. of the ' Meteoroscopium.' The great in-
accuracy of the small canon is also noticed by him ; and it is on this account
that he gives no errata list for it.

Pitiscus, 1613 [T. I.] (pp. 2-271, calculated by Ehetions). Natural
sines for every ten seconds throughout the quadrant, to 15 places, semiqua-
drantally arranged, with first, second, and third differences. (On p. 13, Per-
2}endieulum and Basis are printed instead of Sinus and Sinus complementi).

[T. II.] (pp. 2-61, calculated by Rheticus). Natural sines for every
second from 0° to 1°, and from 89° to 90°, to 15 places, with first and second
differences.

[T. III. and IV.] (pp. 3-15). The lengths of the chords of a few angles,
to 25 places, with verifications &o., followed by natural sines and cosines
for the tenth, twentieth, and fiftieth second in every minute to 35', to 22
places, with first, second, third, fourth, and sometimes fifth differences.

The numbering of the pages thus recommences in each table (except. T.
IV.) ; and each has a separate titlepage. On the first two the date is printed
cIo . lo . xni. instead of do . loc . xiii.

The rescue of the MS. of this work from destruction by Pitiscus (as told by
himself in the preface) forms a striking episode in the history of mathematical
tables. The alterations and emendations in the earlier part of the corrected
edition of the ' Opus Palatinum ' were made by Pitiscus; and he remarked that
a table of sines to more places than ten was requisite to enable the corrections
to be conveniently made. He had his suspicions that Eheticus had himself cal-
culated a ten-second canon of sines to fifteen decimal places; and on application
to Valentine Otho, the original editor of the ' Opus Palatinum,' the latter, who
was then an old man, acknowledged that such was the case, but could not
remember where the MS. was (" ob memoriae senilis debilitatem "). He thought
that perhaps he had left it at Wittemberg ; and accordingly Pitiscus sent a
messenger there to search for it ; but after considerable expense had been in-
curred he returned without it. After the death of Otho, when the MSS. of
liheticus, which had been in his possession, passed into the hands of James
Christmann, the latter discovered the canon among them, when it had been
^iven up for lost. As soon as Pitiscus knew this he examined the MSS. page
by page, although they were in a very bad condition (situ et squalore obsitas
ac psene foetentes), and to his great satisfaction found : — (1) the ten-second
canon of sines to 15 places, with first, second, and third differences (printed
in the work under notice) ; (2) sines for every second of the first and last
degrees of the quadrant, also to 15 places, with first and second differences ;
(3) the commencement of a canon of tangents and secants, to the same
number of decimal places, for every ten seconds, with first and second dif-
ferences ; (4) a complete minute-canon of sines, tangents, and secants, also
to 15 decimal places. From this account, taken in connexion with the
' Opus Palatinum ' and the contents of the present work, one is able to
form some idea of the enormous computations undertaken by Bhetieus;

ON MATHEMATICAI. TABLES. 45

h.is tables not only to this day remain uhsuperseded and the ultimate authori-
ties, but also formed the data whereby Ylacq calculated his logarithmic
canon. Pitiscus says that for twelve years Eheticus constantly had some com-
puters at work (duodecim totos annos semper qliquot Logistas aluit) ; and how
much labour and expense on his p&rt would have been wasted but for the
zeal of Pitiscus is painfal to contemplate ; as it was, it is matter of regret
that Rheticus did not live to sec the publication of either of his canons,
the first of which appeared twenty years, and the other thirty-seven years
after his death. It was Pitiseus's intention to add Eheticus's minute-canon
of tangents and secants ; but they laboured under the same defect as those in
the (uncorrected) ' Opus Palatinum,' and on this account he was dissuaded
from so doing by Adrianus Eomanus. The matter spoken of above as
[T. III. and IV.] was due to Pitiscus himself, and was introduced at the
advice of the same mathematician.

The enormous work undertaken by Eheticus needs no eulogy ; and the
earnestness and love of accuracy displayed by Pitiscus, not only rendered
apparent by his acts but also evident in the prefaces to his several works,
win always render his an honoured name in science.

Tho present work is exceedingly rare ; and the copy we have examined is
in tho library of the Greenwich Observatory. It, the ' Opus Palatinum,'
and ViAca's ' Arithmetiea Logarithmica,' 1628, and ' Trigonometria Artifici-
alis,' 1633, may be said to be the four fundamental tables of the mathemati-
cal sciences.

Gernerth (in the work cited under EnEiiCTrs, 1596, swpra) has given a
list of 88 errors that he detected in the first 7 or 8 places of the canon of
sines; he detected altogether 110; but 22 he states were given by Vega
in his ' Logarithmisch-trigonometrisohe .... Tafeln und Formeln,' Vienna,
1783 : this was Vega's first publication of tables ; and we have not seen the
work.

Grienberger, 1630. Sines, tangents, and secants, to 5 places, for eveiy
minute from 0° to 45° (with foot entries also ; but the table is only half a
complete canon, as e.g. sin 50° could not be taken out from it). There are five
more figures added to the sines, but separated from them by a point (this is
not a true decimal point, as is evident from the rest of the work, whore no
trace of decimals occurs), the object the author had in view in adding them
being that when the sines had to be multiplied by large numbers, the re-
sults might still be correct to the last unit (radius 100,000). Grienberger
stated that more than 35 years before (about 1595) he had calculated a
canon of sines to 16 places, and made considerable progress with the secants
when the ' Opus Palatinum ' appeared and caused him to lay aside his work.
This he regretted exceedingly at the time of writing the present work, as he
was not able to add the five extra figures to the tangents and secants, which
he had transferred from his MS. in the case of the sines. The ' Opus Pala-
tinum' contained enough figures; but some of them were doubtful, and he
wished no doubt to attach to any part of his table. The book is a duodecimo
volume, and would scarcely have been noticed here, but from the fact of part
of it having been the result of an original calculation. ITapier's bones are
mentioned, but not logarithms. The preface contains Grienbcrger's 39-figure
value of TT (see ' Messenger of Mathematics,' July 1873) ; and it was in con-
nexion therewith that we sought the work out, and learnt with some surprise
of Grienbcrger's incomplete and unpublished calculations. The copy we
examined is in the British Museum.

Massaloup, 1847, T. I> The first five hundred multiples of the sines and

46 REPORT— 1873.

cosines of all angles from 1° to 45° at intervals of 10' to two places. The table
occupies 442 closely printed pages.

T. II. gives the first 109 mnltiples of the sine of all angles from 0° to 15°
at intervals of 1' to two places.

The above is- the mathematical description of these tables ; but in the
book, which is intended for surveyors &c., the multiples correspond to differ-
ent lengths (1.0, 1.1,. . . .50.0 Euthen) of the hypothenuse; and the sine
and cosine columns are headed Ebhe and Orundlinie, and are given in
Ruthen. As the arguments are at intervals of a Fuss (= Jjj- of a Euthe)
the table exhibits the results apparently to 3 places. The arrangement in
T. I. is difierent from that in T. II., as while in the former the Euthen and
Fiisse run down the column, and the minutes along the top line (so that aU
the multiples of the same sine or cosine are given consecutively), in T. II. the
minutes run down the column, and the Fiisse along the top line (so that the
same multiples of different angles are given consecutively). In this table also
the results are given to 3 places, if the method of statement used in the book
be followed. As it has been assumed that a Euthe = 10 Fuss, while fre-
quently it = 12 Fuss-, T. III. is given to convert decimals into duo-
decimals, or, more strictly, Euthen Decimalmaass into Werkmaass and
Bergmaass.

T. I. and II. are of course simple traverse tables.

Junge, 1864. Natural sines and cosines for every ten seconds of the
quadrant to 6 places. The table is one of the clearest we have seen, the
figures being distinct, and plenty of space being left between the columns
&c., so as to give a light appearance to the page, though its large size is
rather a disadvantage. The tabular results were interpolated for by Thomas's
calculating machine from the natural sines in HtrissB's tables ; and the last
figure maybe in error by rather more than half a unit. The connexion
between the tables and Thomas's machine, referred to in the title and in the
preface, merely amounts, we suppose, to this — that while computers in
general use log sines, those who possess Thomas's machine will find it
easier to dispense with logaritlims and use natural sines and ordinary
arithmetic.

^'Clouth. Natural sines and cosines (to 6 places) and their first nine
multiples (to 4 places) for every centesimal minute of the quadrant, arranged
scmiquadrantally, the sines and their multiples occupying the left-hand pages,
and the cosines the right ; the arguments are also expressed in sexagesimal
minutes and seconds, the intervals being then 32"-4. We have not seen tho
work itself, but only a prospectus, containing 2 pp. (108 and 109) as specimens.
Judging from this, the book would contain 208 pp. In the cop}' of the pro-
spectus before us, the words " Mayen (chez I'auteur) " are covered by a piece
of paper on which is printed " Halle, Louis Nebert, Libraire-Editeur."
There is no date; but we should judge the tabic to have been only recently
published.

We have also seen advertised ' Tafeln zur Berechming goniometrischer
Co-ordinaten,' by F. M. Clouth — no doubt a German edition of the same
work.

The following is a classified list of trigonometrical tables described in
§4.

Sines, tangents, secants, and versed sines. — (To 7 places) Hantscm, 1827,
T. V. ; WiLLicH, 1853, T. 3; Hriioir, 1858, T. IX,

(To 6 places) Qamkaith, 1827, T, VI;
• Sines, tangents, and secants. — (To Tplaces) Sir J. Moore, 1681 [T. III.] ;

ON MATHEMATICAL TABLES. 47

ViAca, 1681 [T. I.] ; OzANAM, 1685 ; Sherwin, 1741 [T. IV.] ; Hsnt-
R0H15N (ViACQ), 1757 [T. I.] ; ScHULZE, 1778 [T. V.] ; Lambert, 1798, T.
XXYi. ; Dor&iAs, 1809 [T. III.].

(To 6 places) Otjghtred, 1657 [T. I.] (centesimal division of the degree) ;
Uesinus, 1827 [T. V.] ; Bbaedmobe, 1862, T. 38.

(To 5 places) HoUel, 1858, T. II. ; Peieks, 1871 [T. V.].

Sines and tangents (only). — (To 7 places) Bates, ,1781 [T. II.]; Veaa,
1797, T. III. ; HoBBKT and Ibeleu, 1799 [T. I.] (centesimal) and B (cen-
tesimal) ; (?) *Sai,omon, 1827, T. XII.; Tdekish LoaABiinMs (1834];
HtTLssE's Vega, 1840, T. III.

(To 6 places) Tbottee, 1841 [T. IV.].

■ (To 5 places) Schuidi, 1821 [T. III.]; Rankine, 1866, T. 6; Wackee-
BAETH, 1867, T. VIII.

(To less than 5 places) Paekhtosi, 1871, T. XXX. and XXXI.

Tangents and secants (only). — Dokn, 1789, T. V. (4 places) ; [Seeep-
SHAM-KS, 1844] [T. IV.]: (4 places).

Sines (alone). — (To 15 places) Caiiet, 1853 [T. VII.] (centesimal).

(To 7 places) DoNir, 1789, T. Ill; Hassieb, 1880 [T. V.].

(To 6 places) Maskeltne (Requisite Tables, Appendix), 1802, T. I.; Dxrcoit,
1820, T. XIX. ; Kbeigan, 1821, T. IX.; J. Taylob, 1833, T. XX.; Nobie,
1836, T. XXVI. ; Geifein, 1843, T. 19 ; J. Tayloe, 1843, T. 32 ; Domkb,
1852, T. XXXVI.

(To 5 places) Lambebt, 1798, T. XXV. ; Maskblyne (Requisite Tables),
1802, T. XVII. ; Bowbitoh, 1802, T. XIV. ; Mooee, 1814, T. XXIV, ;
Waliace, 1815 [T. III.] ; Geegoey, &c., 1843, T. X,

Multiples of sines.— ScsvLzs, 1778 [T. VI.] ; Lambebt, 1798, T. XXV.

Versed sines (done).— {To 7 places) Sir J. Mooee, 1681 [T. IV.] ; [Sir
J. Mooee, 1681, Versed sines'] ; Dodson, 1747,. T. XXVI. ; Dougias, 1809,
[T. IV.] ; Faeley, 1856 [T. I.].

(To 6 places) Maskelyne (Requisite Tables, Appendix), 1802, T. II.;
Mackay, 1810, T. XLI. ; Lax, 1821, T. XVII. (and .coversed &c. sines).;
RiDBiE, 1824, T. XXVIII. ; Noeie, 1836, T. XXXVI. ; RtinKEE, 1844;
T. III. ; Inman, 1871 [T. VIII.] and [T. IX.].

■ Sines &c. expressed in radicals. — Lambebt, 1798, T. XIX. ; Uesinus,
1827 [T. III.]; Vega, 1797, Appendix.

Miscellaneous. — Sin^ ^, Anbeew, 1805, T. XIII ; siu^ x and tan= iv,

Pasquich, 1817, T. II. ; suversed, coversed, sucoversed sines, Lax, 1821, T.
XVII. ; I sin ic, Stansbuey, 1822, T. Y; sexagesimal cosecants and cotan-
gents, Bevebley (1833 ?), T. VI. (pp. 232 &c.) ; sexagesimal sines. Id. T.

XV.; sin ?,Hti,ssB'sVESAT.IV.1840;sin^^', [Sheepshanks, 1844] [T. VI.] ;

2 ^

i sin X expressed sexagesimally, Goebon-, 1849, T. XVIII. ; see also Schio-

Mn,CH[1865?].

j}fote. A list of tables in which both natural and logarithmic functions are

given side by side in the same table is added at the end of § 3, art. 15.

Art. 11. LengtTis of Circular Arcs.

Tables of the lengths (or longitudes) of circular arcs are very frequently
given in collections of logarithmic aiid other tables ; but we have seen none
of snfficient extent to be published separately. Angles are measured either
by degrees, minutes, &c., or by the ratio which the corresponding arc bears

48 REPORT— 1873.

•to the unit arc, or arc equal iu length to radius. The latter method is usually
described in English text-books under the title "Circular Measure;" so that
in the descriptions in § 4 we have spoken indifferently of the length of the •
arc of x°, the longitude of aP, or the circular measure of of. The tables of
circular arcs usually give the circular measure of 1°, 2°,'. . up to 90°, 180'^,

or sometimes 360°, of 1', 2', 60', of 1", 2", 60", and very often of

1'", 2'", 60'" also. By means of such a table any number of degrees,

minutes, &c. can be readily expressed in circular measure.

The following is a detailed list of the hngtlis of circular arcs contained in
works described in § 4 : —

(To 44 places) Hobebt and Idelee, 1799, G (centesimal division).

(To 27 places) Acad£mie de Pbusse, 1776 [T. II.] ; Schttlze, 1778
[T. VII.]; Lamheet, 1798, T. XXIIT.

(To 25 places) Callbi, 1853 [T. V.] (sexagesimal and centesimal).
- (To 15 places) Hantscul, 1827, T. X.

(To 12 places) Schmidt, 1821 [T. IV.] ; M&llek, 1844 [T. IV.].

(To 11 places) Vega, 1794, T. II.; HtJLSSE's Vega, 1840, T. II.; KoniEn,
1848 [T. V.].

(To 10 places) Shoetrede, 1849, T. III. ; Beuhns, 1870.

(To 8 places) Vega, 1797, T. III. ; Peaesok, 1624 [T. III.].

(To 7 places) Dodsow, 1747, T. XXV. ; Ursiots, 1827 [T. III.] ; Gei;-
soN, 1832, T. VI.; Trotter, 1841 [T. VII.] ; Shoetrede (tables), 1844,
T. XXXVIII. ; Waenstorfp's Schumacher, 1845 [T. II.] ; Wilkch, 1853,
T. D ; Beemiker's Vega, 1857, T. II. ; Hutton," 1858, T. XI. ; Dupttis,
1868, T. IX. ; Peters, 1871 [T. III.]

(To 6 places) Beemieer, 1852, T. II.

(To 5 places) Wackeedarth, 1867, T. IV.

See also Vega, 1800, T. II.; Byrne, 1849 [T. II.] ; *Schi,omhoh
[1865?].

Art. 12. Tables for (he expression of JiOiirs, minutes, ^e. as decimals of a day,
and for the conversion of time into sj)ace, and vice versd.

The largest table we have seen to convert hours, minutes, &c. into decimals
of a day is HoUel, 1866. Tables of this kind are not numerous.

Three hundred and sixty degrees of space or arc are equivalent to twenty-
four hours of time ; so that 1" corresponds to 15°, 1"° to 15', and 1" to 1 5" ;
1" is therefore 4 thirds of time = 4* ; 36' =2°' 24' &c. Small tables to convert
space (arc, or longitude) into time are not unfrequently given in collections
(generally nautical) of tables. A complete table of the kind gives the numbers
of hours and minutes corresponding to 1°, 2°, . . . . 360° ; and the same figures
also denote the number of minutes and seconds, and seconds and thirds (of

time) corresponding to 1', 2' 360', or 1", 2", . . . .360" respectively. In

this Eeport "", "", ", &c. are used to denote hours, minutes, seconds, and thirds (of
time), and °, ', ", '" for degrees, minutes, &c. of space — a distinction which it
is often convenient to adopt.

Iiittrow, 1837. T. I.-IV. (5 pp.) are small tables for the conversion of are
into time &c. All the other tables, which occupy more than nine tentlis of
the tract, are astronomical.

Hoiiel, 1866 (Time Tables), T. IT. To convert hours, minutes, and
seconds into the decimal of a day (pp. 15). Any number of hours, minutes,
and Beconds (and fractions of a second, as proportional parts are added)

ON MATHEMATICAL TABLES. 49

can be readily expressed as a decimal (to seven places) of a day, and vice
versdhj means of it.

The following are tables described in § 4 : —

Tables for the conversion of Time into Space, and vice versd. — Cboss-
yrELL, 1791, T, XIII.; Bowdixoh, 1802, T. XII.; Eios, 1809, T. XVI.;
Kekigan, 1821, T. XIII. ; Stansbubt, 1822, T. I. ; PEAKsoisr, 1824 [T. I.] ;
Galbraixh, 1827, T. XII. (Introd.); 'WAitNSTOKFp's Schumachee, 1845 [T. I.];
KoHLEE, 1848 [T. I.] ; GoEDoif, 1849, T. XI. ; Domke, 1852, T. XLVII. and
XLVIII. ; Bebmikbb, 1852, T. II. ; Thomson, 1852, T. I. ; Beemikbe's Vega,
1857, T. III. ; Hotel,, 1858, T. I. ; Petees, 1871 [T. II.].

Tables to express Degrees, Minutes, 4rc. as decimals of a right angle,
or Hours, Minutes Sfc. as decimals of a day, and vice versd, Sfc. — Hobeet
andlDELEE, 1799, C. I.-IV., D. I.-III., E. I.JV., F. ; Galbkaith, ]827,
T. XI. (Intrpd.); Hantschi,, 1827, T. XTI. ; BEvEELsr (1833?), T. VI.
(p. 127) ; KdHLEE, 1848, T. IX. ; Petees, 1871 [T. I.].

Art. 13. Tables of (Briggian) Logarithms of Numbers.

The facts relating to the invention of Briggian (or decimal) logarithms are
-as follows :— In 1614 Napibe published his ' Canon Mirificus ' (see § 3,
art. 17), which contained the first announcement of thejnvention of logarithms,
and also a table of logarithmic sines, calculated so as to be very similar to what
are now called hyperbolic logarithms. Heket BEiees, then Professor of Geo-
jnetry at Gresham College, London, and afterwards Savilian Professor of Geo-
•metry at Oxford, admired this work so much that he resolved to visit Napier.
■" Naper, lord of Markinston, hath set my head and hands at work with his
new and admirable logarithms. I hope to see him this summer, if it please
God ; for I never saw a book which pleased me better, and made me moro
■wonder." This he says in a letter to Usher (Usher's ' Letters,' p. 36, accord-
ing to Ward). Briggs accordingly visited Napier, and stayed with him a
whole month (in 1615). He brought with him some calculations he had
made, and suggested to Napier the advantages that would resultfrom the choice
«f 10 as a base, having publicly explained them previously in his lectures
at Gresham College, and written to Napier on the subject. Napier said that
he had already thought of the change, and pointed out a slight improvement,
viz. that the characteristics of numbers greater than unity should be posi-
tive, and not negative, as Briggs suggested. Briggs visited Napier again in
1616, and showed him the work he had accomplished, and, as he himself says^
would have gladly paid a third visit in 1617, had Napier's life been spared
(he died April 4, 1617). The work alluded to is Bbigss's ' Logarithmorum
■Chilias Prima,' which was' published (privately, we believe) in 1617, after
Napier's death, as in the short preface he states that why his logarithms are dif-
ferent from those introduced by Napier "sperandum, ejuslibrumposthumum,
abunde nobis propediem satisfaoturum." The liber posthumus was Napier's
« (Donstructio,' which appeared in 1619, edited by his son (see § 3, art. 17),
Brings continued to labour assiduously, and in 1624 published his 'Arith-
metiea Logarithmica,' giving the logarithms of the numbers from 1 to
20,000, and from 90,000 to 100,000 (and in some copies to 101,000), to 14
places.

To the above facts we must add that Napier made a remart, both in Wright's
translation of the ' Descriptio ' (1616) and in the ' Eabdologia' (1617), to the
«£fect that he intended in a second edition to make an alteration equivalent
io taking the logarithm of 10 equal to unity.

We have thought it proper to give the circumstances attending the inveur
1873. E

50 REPORT— 1873.

tiou of Briggian logarithms in the above detail, as there seems every ^roba-
bUity that the relations of Napier and Briggs may beccane a subject of con-
troversy among those who have never taken the trouble to examine the
■original sources of information. Hutton, in his ' History of Logarithms '
•(prefixed to all the early editions of his logarithmic tables, and also printed
in vol. i. pp. 306-340 of .his • Tracts,' 1812), has unfortunately interpreted all
Briggs's statements with regard to the invention of decimal logarithms' in a
manner clearly contrary to their true meaning, and unfair to Napier. In
reference to the remark in Briggs's preface to the ' Chilias,' that it is to he
lioped that the posthumous work will explain why the logarithms are difiPerent
from Napier's, Hutton proceeds : — "And as Napier, after communication had
with Briggs on the subject of altering the scale of logarithms, had given notice,
both in Wright's translation and in his own ' Eabdologia,' printed in 1617,
of his intention to alter the scale (though it appears very plainly that he never
intended to compute .any more), without making any mention of the share
which Briggs had in the alteration, this gentleman modestly gave the above
hint. But not finding any regard paid to it in the said posthumous work,
published by Lord Napier's son in 1619, where the alteration is again adverted
to, but stUl without any mention of Briggs, this gentleman thought he could
not do less than state the grounds of that alteration himself.

" Thus, upon the whole matter, it seems evident that Briggs, whether he had
thought of this improvement in the construction of logarithms, of making I
the logarithm of the ratio 10 to 1 before Lord Napier or not (which is a secret
that could be known only to Napier himself), was the first person who com-
municated the idea of such an improvement to the world ; and that he did
this in his lectures to his auditors at Gresham College in the year 1615, very
soon after his perusalof Napier's ' Canon Mirificus Logarithmorum ' in the year
1614. He also mentionedit to Napier, both by letter in the same yearapdonhis
first visit to him in Scotland in the summer of the year 1616, when Napier ap*
proved the idea, and said it had already occurred to himself, and that he had
determined to adopt it. It would therefore have been more candid in Lord
Napier to have told the world, in the second edition of this book, that Mr»
Briggs had mentioned this improvement to him, and that he bad thereby been
confirmed in the resolution he had already taken, before Mr. Briggs's com-
munication with him, to adopt it in that his second edition, as being better
fitted to the decimal notation of arithmetic which was in general use. Such
a declaration would have been but an act of justice to Mr. Briggs ; and the
not having made it cannot but incline us to suspect that Lord Napier was
desirous that the world should ascribe to him alone the merit of this very
useful improvement of the logarithms, as well as that of having originally in-
vented them ; though, if the having first communicated an invention to the
world be sufficient to entitle a man to the honour of having first invented it,
Mr. Briggs had the better title to be called the first inventor of this happy
improvement of logarithms."

The above comments of Hutton's are all the more unfortunate because they
occur in a history that is generally accurate and truthful. It is needless
to say that, the facts being as above narrated, there is not the smallest
ground for imputing unfairness to Napier ; but Hutton seems to have some-
how become possessed of such an idea and read all the facts by the light of it.
On the other hand, however, some of the accounts are scarcely fair to Briggs'.
Mr. Mark Napier, in his ' Memoirs of John Napier,' has successfully refuted
Hutton ; but he has fallen into the opposite extreme of extravagantly eulogizing
Napier at the expense of Briggs, whom he redupes.to the level, of a mere

ON MATHEMATICAL TABLES. 51

computer ; and in these terms Mr. Sang lias algo recently spoken of the latter,
Mr. Napier attributes Hutton's assertions to national jealousy (!) ; and it wili
be a matter of regret if any other writers should follow his example in at-
tempting to glorify Napier by depreciating Briggs. The words of the latter;
in the 1631 translation (and amplification, see below) of his ' Arithmetioa ' of
1624, are :— « These numbers were first invented by the most excellent John
Neper, Baron of Marchiston ; and the same were transformed, and the founda-
tion and use of them illustrated with his approbation [ex ejusdem sententia]
by Henry Briggs." No doubt the invention of decimal logarithms occurred
to both Napier and Briggs independently ; but the latter not only first an-
nounced the advantage of the change, but actually completed tables of the
new logarithms. Thus, as regards the idea of the change, Napier and
Briggs dinde the honour equally ; whUe, on the principle that "great points
belong to those who make great points of them," nearly aU belongs to Briggs.

On the subject of Briggs and the invention of logarithms, see the careful
and impartial life of Briggs in "Ward's ' Lives of the Professors of Gresham
College,' London, 1740, pp. 120-129, and also ' Yits quorundam eruditis-
simorum et iUustrium virorum' &c., scriptore Thoma Smitho, Londini, 1707
(Vita Henrici Briggii), as well as ' Memoirs of John Napier of MercMston,' by
Mr. Mark Napier, Edinburgh, 1839, and the same author's ' Naperi libri qui
supersunt' (see § 3, art. 17). See also Hutton's account (reference given above)
and Phil. Mag., October and December (Supplementary No.) 1872, and May
1873. It is proper to add that the date we have given for Briggs's first visit
to Napier, viz. 1616, is difi'erent from that assumed by other writers, viz. 1616 ;
we have, however, little doubt that the former is correct, as it in all respects
agrees with the facts. The reason that "Ward, Hutton^ &c. assign Briggs's
first visit to 1616, and the publication of the ' Chilias' to 1618, is, no doubt,
due to the fact that they supposed Napier to have died in 1618 ; but Mr. Mark
Napier has shown that the true date is 1617 ; and this brings all the facts into
agreement (see Phil. Mag. December 1872, Supp.).

, Like Napier, Briggs was not very particnlar about the spelling of his name.
In Wright's translation it appears as Brigs on the titlepage,,Brigges on th?
first page of the prefect, and Briggs in the eulogistic verses.

Although we have spoken of logarithms to the base 10 &c., we need scarcely
observe that, although exponents and even fractional exponents were in a sort
of way introduced by Stevinus, neither Napier nor Briggs, nor any one till
long after, had any idea of connecting logarithms with exponents.

To return to the original calculation of the logarithms of numbers. Briggs,
as has been stated, published the logarithms of the numbers from 1 to
20,000 and from 90,000 to 100,000 to fourteen places, in his ' Arithmetica.'
There was thus left a gap from 20,000 to 90,000, which was filled up by-
Adrian Ylacq (although Briggs had in the mean time nearly completed th^
necessary calculations ; see Phil. Mag. May 1873), who published at Gouda,
in 1628, a table containing the logarithms of the numbers from unity t9
100,000 to 10 places of decimals. Having calculated 70,000 logarithms and
copied only 30,000, "Vlacq would have been quite entitled to have called his
a new work. He designates it, however, only a second edition of Briggs,
the tide running, "Arithmetica logarithmica sive logarithmorum chiliades

centum, pro numeris natural! serie crescentibus ab Unitate ad 100000

Editio secunda aucta per Adrianum Vlacq, Goudanum Goudse, excitdebat

Petrus.Eammasenius. 1628." This table of Vlacq's was published, with an
English explanation prefixed, in London in 1631, under the title, " Logarith-
saicaE Arithmetike, or Tables of Logaritbmes for absolute numbers, from aa,

322

52 REPORT— 1873.

unite to 100000 London, printed by George MiUer, 1631" (fuU tiUes are

given in § 5). '

Speaking of Briggs's ' Arithmetica Xogarithmica ' of 1624, De Morgan, in
his article on Tables in the ' English Cyclopaedia,' says :— " After his [Briggs's]
death, in 1631, a reprint was, it is said, made by one George Miller ; the
Latin title and explanatory parts were replaced by English ones — ' Logarith-
micall Arithmetilce ' &e. We much doubt the reprint of the tables, and think
that they were Briggs's own tables, with an English explanation prefixed in
place of the Latin one. Wilson (in his ' History of Navigation,' prefixed to
the third edition of Eobertson) says that some copies of Vlacq, of 1628, were
purchased by our booksellers, and published at London with an English ex-
planation premised, dated 163] . Mr. Babbage (to whose large and rare col-
lection of tables we were much indebted in the original article) has one_ of
these copies ; and the Enghsh explanation and title is the same as that which
was in the same year attached to the asserted reprint of Briggs. We have no
doubt that Briggs and Vlacq were served exactly in the same manner." On
referring to Eobertson (fourth edition, p. xvi), there is found to be no further
information than that contained in the above extract. That De Morgan's
suggestion is quite correct, and that Miller's and Vlacq's tables are both
printed from the same types, we have assured ourselves by a most careful
comparison, which leaves no doubt whatever that the two works are printed
from the same type throughout. We are thus enabled to state that the
same errata-list suffices for both ; and this is important, as VLAca (1628,
or 1631) is still the most convenient and most used ten-figure table in ex-
istence. Briggs's friends were annoyed at Ylaeq's publication ; but it must
be borne in mind that their objections have reference, not so much to the table
(which is the only thing of practical importance now) as to the prefixed tri-
gonometry, which Vlacq curtailed in his " second edition." George Miller also
published some copies of the original 'Arithmetica' of 1624, with the same title-
page and introduction as were prefixed to the copies of Vlacq of 1628 ; and this
was distinctly wrong, as the titlepage does not in this case describe the con-
tents correctly.

It thus appears that Beiggs's table was published in 1624, and VLAca's in
1628 — that copies of the tabular portions of both these works were obtained by
George MiUer, and published by him in 1631, with the same (English) title-
page and introduction, which, though correctly describing the contents of
Vlacq, is quite inappropriate for Briggs. This has led to a very great amount
of confusion, which has been greatly increased by the fact that on the title-
pages Briggs's and Neper's names occur, and that 'Vlacq only called his work
a second edition. It is in consequence exceedingly common to see Vlacq's
work assigned to Briggs or Neper ; and it is almost invariably ascribed to one
or other of the latter in the catalogues of libraries.

ViAca's 'Arithmetica' of 1628 was also published with the same date, with
a, French title (" Arithm^tique Logarithmetique " &c.) and introduction.
Vlacq modestly describes his share of the calculatiott in the words : — " La
description est traduit du Latin en Frangois, la premiere Table augment^e,
«t la seconde composee par Adriaen Vlacq." Miller's (1631) copies of Vlacq
are not so rare as the extract from De Morgan might imply. We have seen
five of them, and only three or four of the original (1628) works (including
both Latin and French).

In 1631 VLAca published his ' Trigonometria Artificialis' (§4). This
■Work contains, among other tables, the logarithms of the numbers from unity
to 20,000, printed also (with the exception of the last sheet, referred to fur-
ther on) from the same type.

ON MATHEMATICAL TABLES. .53

No further calculation of logarithms of numbers took place tiU the end of
the last century, when the great French manuscript tables (the ' Tables
BTj Cabastkb ' — see description of them below) were computed under the
direction of Prony. These, as is well known, have never been published.

In 1794 Veoa published his ' Thesaurus Logarithmorum Completus,' which
contains a complete ten-figure table from 1000 to 101,000. It was not, how-
ever, the result of a fresh' calculation, but was copied from Vlacq, after ex-
amination and correction of many errors (see Vesa's ' Thesaurus,' § 4).

In 1871 Mr. Sang published his seven-figure table of logarithms of numbers
to 200,000, the second half of which was obtained by a new calculation. It is
thus seen that, with the exception of the Tables hv Cadasiee, and the second
half of Mr. Sang's table, every one of the hundreds of the tables that have
appeared has been copied from Briggs or VLAca ; and considering the enor-
mous number of calculations in which logarithms have been employed,
and the vast saving they have effected of labour, it must be admitted that
(apart from the fact that the great tables of Beiggs and YLAca remain
still unsuperseded) great historical interest attaches to the original com-
putation.

ViiAca's ten-figure table contains about 300 errors (leaving out of consi-
deration errors affecting only the last figure by a unit). The greater number
of these were found either by Vega, or by Lefort from comparison with the
Tables dv Cadastre : complete references and a smaiU subsidiary list are
given in the ' Monthly Notices of the Eoyal Astronomical Society '- for May
and June 1872. "While speaking of ten-figure logarithms, we may men-
tion PiNEio's table described below ; but VLAca (1628 or 1631) and Vega
(1794) are far preferable : they are unfortunately so rare, however, that not
many besides those who have access to a good library can make use of
them, and, except to a few, the employment of ten-figure logarithms in their
most convenient form is denied : we much prefer Vlacq to Vega for use, as
the arrangement is more convenient.

To return to the history of logarithmic tables to a less number of figures.
In 1625 Wingate published at Paris his ' Arithmetique Logarithmetique,' con-
taining seven-figure logarithms to 1000, and logarithmic sines and tangents
from GuNTEE (see De Morgan ; the fuU title of the Gouda edition of Wingate
(1628) is given by Hogg, p. 408), thus introducing Briggian logarithms into
Prance ; and in 1626 appeared both Heneion's 'Traicte' (§ 4) at Paris, con-
taining 20,000 logarithms from Briggs and Gunter's logarithmic sines and
tangents, and Db Decker's ' Nieuwe Telkonst ' (§ 4) at Gouda, giving also
logarithms from Briggs and Gunter; then Vlacq began to calculate logarithms,
and brought them in 1628 to the state in which they now are.^ There is a table
of logarithms in Norwood's ' Trigonometric' (1631) ; and in 1633 appeared
Roe's table (§ 4), in which the first four figures of the logarithm are printed
at the top of the column . This was an advance halfway to the modern arrange-
ment, which was introduced in its present form in John Newton's eight-figure
table'(1658). On PArLHABER, 1631, and Oughtred, 1657, see § 4. , _

Tables of seven- and five-figure logarithms are too numerous to notice
here separately. The chief line of descent is Briggs, Vlacq, Roe, perhaps
Newton, the editions of Sheewin, Gaedinee; and then both Htttion and
Callet bring down the succession to the present day. A very fair account
of several logarithmic tables is given by Eogg in section iv. " Elementar-
Geometrie " (B) of his ' Handbuch,' who *dded to the books described m this
part of his bibliography a description of the contents. Bat the reader must
be warned against trusting his accounts, except where he is clearly describijig

54 REPOKT — ] 873.

■works he lias seen. Of seven-figure tables we have found Babbage as con-
•venient as any, and it is nearly free from error ; Caixet and Hxttton are also
much used; Shoktkbde and Sang are both conspicuous for giving the multiples
of the differences instead of proportional parts ; the latter work also extends
to 200,000 instead of 100,000 as usual. Of five-figure tables De Moegan's
■(Useful-Ejiowledge Society) tables are considered the best, and are practically
■free from error. We cannot, however, here particularize the advantages of
the different tables, which must be gathered from their full descriptions.
Some of them have, of course, been merely included on account of their his-
torical value. "We may here mention that the subject of errors in these tables
will be considered in a subsequent Report.

Vega (p. iii of the Introduction to the « Thesaurus,' 1794) says that "Vlacq's
■* Arithmetiea ' (1628) and ' Trigonometria ' (1633) were printed at Pekin in
1721, under the title " Magnus Canon Logarithmorum, turn pro sinibus ac
tangentibus ad singula dena secunda, turn pro numeris absolutis ab imitate ad
100,000. Typis sinensibus in Aula Pekinensi, jussu Imperatoris excusus,
1721 " (three volumes folio, on Chinese paper), and that a copy had been
offered him for sale two years previously (1792), Montucla (' Histoire,'
vol. iii. p. 358) says, the name of the Emperor in question was Kang-hi.

Eogg also (p. 408) confirms Vega, extracting the title from Brunet's
• Manuel du Libraire.'

In the preface to his tables (1849) Mr. FilipowsH concludes by a sneering
remark on the Chinese, saying that Mr. Babbage proved, " as had long been
suspected, from what source those original inventors had derived their
. logarithms ; " and we have noticed this tendency to ridicule the Chinese in
this matter as detected plagiarists in others. In point of fact there is no more
plagiarism than when Babbage or Callet publishes a table of logarithms with-
out the name of Vlacq on the titlepage. The first pubUoation in China, we
infer from Rogg, merely professed to be a reprint of Vlacq ; and if logarithms
came into general use, it is natural that they would be published, as with us,
without the original calculator's name. The fault is with those who form
preconceived opinions on subjects they have not investigated.

A Turkish table of logarithms is described in § 4. A small table of
logarithms to base 2 is noticed below, under Montfebkibr, 1840.

We may mention a little book, ' Instruction ^lementaire et pratique eur
I'usage des Tables de Logarithmes,' by Prony (Paris, 1834, 12mo), which
explains the manner of using of tables of logarithms &c., adapted to Callet.

In many seven-figure tables of logarithms of numbers the values of S and T
are given at the top of each page, with V, the variation of each, for the purpose

of deducing log sines and tangents. S and T are the values of log , and

tan a; *

log for the number of seconds denoted by certain numbers (sometimes

only the first, sometimes every tenth) in the number-column on each page.
Thus, in Callet, 1853, on the page of which the first number is 67200,
„ , sin 6720" , _, , tan 6720" ... ,, _, ,,

S=log „„„„ — and T=log — Wt^ — > while the V s are the variations of

each for 10". To find then, say, log sin 1° 52' 12"-7, or log sin 6732"-7, we
have 8=4-6854980, and log 6732-7=3'8281893, whence, by addition, we
have 8-5136873; but V for 10" is -2-29 ; whence the variation for 12"-7
is —3, and the log sine required is 8-5136870. Tables of S and T are fre-
quently called, after their inventor, Delambre's tables.
;. It is only, since the completioi;! of this Report, and therefore too late to

ON MATHEMATICAL TABLES. 55

make any use of it, that we have received from Professor Bierens de Haan a
copy of a very valuable tract, ' Jeta over Logarithmentafels,' extracted from,
the .' Verslagen - en Mededoelingen. der Koninklijke Akademie van Weten-
sehappen, Afdeeling Natuurkunde,' Deel-xiv. Amsterdam, 1862, 8vo (pp, 80),.
which cotitains by far the most complete list of authors or editors of loga-
rithtaift tables of aU kinds, with the dates and places of publication (from 1614
to 1862), that we have seen, and must be nearly perfect. Some remarks are
made on those of them that de Haan has examined himself ; and there is ap-
pended a vahiable index of reference to papers on logarithms that have ap-
peared in any Journal or Society's Proceedings.

We may also refer to the paper of Gernerth's noticed under Eheiicus,
1598 (§ 3, art. 10), which contains a number of last-flgTire errors in logarith-
mic and other tables. Gemerth was desirous of ascertaining the care bestowed
on the editing of mathematical tables, and considering that it was best
measured by the accuracy of the last figure, he confined himself to the exa-
mination of this point alone (except in the cases of Ehetictjs and Pitisotts,
wiiere the first seven or eight figures were included), and detected very many
errors. He altogether examined tables by eighteen authors ; but generally,
where the errors were numerous, ho has given only five per cent, of those that
he found.

Also, as this sheet is passing through the press, we add references to two
papers in the 'Monthly Notices of the Eoyal Astronomical Society' for
April and May, 1873, " On the Progress to accuracy of Logarithmic Tables,"
and " On Logarithmic Tables ;" in the former of which the number of Ylaeq's
original errors that were reproduced in succeeding works is discussed, while
the latter contains remarks on logarithmic tables both of numbers and trigo-
nometrical fnnctions. An abstract of the first appears also in the ' Journal
of the Institute of Actuaries,' vol. xvii. pp. 352-354.

Briggs, 1617. Logarithms of numbers from unity to 1000 to 14 places
of decimals. This was the first table of Briggian logarithms calculated^ or
published. Neither author's name nor date nor place appears on the title-
page of the work, which is a mere tract of 16 pp. (at all events in the Brit.
Mus. copy) ; but that it was published by Briggs in 1617 is beyond doubt
(see « PhQ. Mag.' he. cit. below).

The preface concludes with the motto "Intenui; sed non tenuis fructusve
laborve." On the work see the introductory remarks to this Article, and
also ' Phil. Mag.' December (Supplementary No.) 1872. It is stated by
Hutton and all' the other writers to be an 8-place table ; but it really is as
described above. One reason for the universal error is that copies are so
extremely rare that we have only been able to see one *, viz. that in the British
Museum, in the catalogue of which it is entered under Logarithms, and
marked as of [1695 ?]. The book is not in the printed Bodleian Catalogue. It
is peculiarly interesting as being the first pubKcation of decimal logarithms.
Nearly all the descriptions and bibliographies wiU be found very erroneous,
several confounding it with Wright's translation of Napier's ' Canon' (see

Briggs, 1624, Logarithms of numbers from 1 to 20,000, and from 90,000
to 100 000 to 14 places, with interscript differences. The characteristics to
the logarithms are given ; and this has led to the table being sometimes erro-
neously described as being to 15 places. The table occupies 300 pages.

* We think wa remember to have met with another among the Birch MSS. in the
British Museum. ,

56: KEPOKT— 1873.

Several lists of errata in this work have been givon-^viz. by Vlacci
in his ' Arithmctica,' by Sheewin in his tables, by Ysni. (folio, 1794), by
Lefoet (' Annales de rObservatoire do Paris'). The introduction oocujries
88 pages, and is in Latin.

- In some copies there is an additional chiliad added, so that the range of
the second portion of the table is from 90,000 to 101,000 ; and there is a
table of square roots of numbers up to 200, to 10 places, occupying the last
two pages : these copies are very rare. There is one in the Library of
Trinity College, Cambridge, with the following note in it by Dr. Brinkley : —

" This is a very scarce copy, having an addition very rarely to be met
with. Vide Hutton's preface to his ' Logarithms,' p. 33, who could never
find a copy with the addition." Mr. Merrifield has also one of these
copies.

On this work see the introductory remarks to this Article.

Tables du Cadastre. On the proposition of Camot, Prieur, and Brunei,
the French Government decided in 1784 that new tables of sines, tangents,
&c., and their logarithms, should be calculated in relation to the centesimal
division of the quadrant. Prony was charged with the direction of the work,
and was expressly required " non seulement k composer des Tables qui ne lais-
sassent rien a desirer quant a I'exactitude, mais Jl en faire le monument de calcul
le plusvaste et le plus imposant qui eut jamais ete execute outoSmecongu," —
an order faithfully carried out. Prony divided the calculators &c. into three
sections : the first consisted of five or six mathematicians (including Legendre),
who were engaged in the purely analytical work, or the calculation of the
fundamental numbers ; the second section consisted of seven or eight calcu-
lators possessing some mathematical knowledge ; and the third comprised
the ordinary computers, 70 or 80 in number. The work, which was done
wholly in duplicate, and independently by the two divisions of computers,
occupied two years.

As a consequence of the double calculation, there are two manuscripts in
existence, one of which has been long deposited in the Archives of the Obser-
vatory ; the other, though supposed to be in the Archives of the Bureau des
Longitudes, was in reality in the possession of Prony's heirs, by whom it was
presented to the Library of the Institute in 1858.

Each of the two manuscripts consists essentially of 17 large folio volumes,
the contents being as follows : —

Logarithms of numbers to 200,000 8 vols.

Natural sines 1 vol.

Logarithms of the ratios of arcs to sines from O'-OOOOO to 1 . ,
©"•OSOOO, and log sines throughout the quadrant . . J ^^ ^"

Logarithms of the ratios of arcs to tangents from O'-OOOOO 1

to 01-05000, and log tangents throughout .'the 1-4 „
quadrant J

It would take too much space to state the intervals &c. in detail. Speaking
generally, the trigonometrical functions aregivenfor every hundred-thousandth
of the quadrant (10" centesimal or 3" -24 sexagesimal). The tables were all
calculated to 14 places, with the intention of publishing only 12 ; but M. Le-
fort, who has recently examined them, states that the twelfth figiu-e may bo in
error by as much as 0-8 of a unit in this place, though a little additional care
would have rendered it more accurate. The Institute copy has also a table of the
first 500 multiples of certain sines and cosines ; and the Observatory copies
have an introduction containing, among several other subsidiary tables, the first

ON MATHEMATICAL TABLES. 57

26 powers of ^ to 28 figures. It may be mentioned that tho logarithms of

10,000 primes -were calculated to 19 places, and the natural sines for every
minute (centesimal) to 22 places. This account of the ' Tables du Cadastre '
has been abridged from a memoir by M. Lefort, in t. iv. (pp. [123]-[150]) of
the ' Annales de I'Observatoire de Paris ' (1858), where an explanation of the
methods of calculation, with the formulaj &c., is given. The printing of the
table of natural sines was once begun. M. Lefort says that he has seen six
copies, all incomplete, although including the last page. De Morgan also men-
tions that he had seen some of th^ proofs. Babbage compared his table with
the ' Tables du Cadastre ;' and M. Lefort has given, by means of them, most
important lists of errors in Vlacq and Beiggs ; but these are almost the only
uses that have been made of tables the calculation of which' required so great
an expenditure of time and money. " In 1820," says De Morgan, " a dis-
tinguished member of the Board of Longitude, London, was instructed by our
Government to propose to the Board of Longitude of Paris to print an abridg-
ment of these tables, at the joint expense of the two countries. £5000 was
named as the sum which our Government was willing to advance for this
purpose ; but the proposal was declined " (Penny Cyclopaedia, Article
" Prony "). The value of the logarithms of numbers is now materially
lessened by Mr. Sang's seven-figure table from 20,000 to 200,000 (see
San&, 1871, in this Article).

liogg (p. 241) gives the title " Notice siir les grandes tables logarithm, et'
trigonom. calculees au Bureau du Cadastre," Paris, an IX. (=1801), and
on the subject gives a reference to Benzcnberg's ' Angewandte Geom.' iii.
p. 557.

Hill, 1799. Five-figure logarithms from 1 to 100 and from 1000 to
10,000, printed at fuU length, and with characteristics — no differences
(pp. 23-38). The author was an accountant; and the table was intended
for commercial purposes, its use in which is explained in the book,

Reishammer, 1800. These are commercial logarithms, intended for
merchants &c. When the number is less than unity, the logarithm of its
reciprocal (which the author calls the logarithme negatif) is tabulated; if
greater than unity, its own logarithm (Jogarithme positif). _ The first table
(which only occupies 2 pages) gives the logarithmes nSgatifs of the frac-
tions from -^ to 1, at intervals of -^-^ to 5 places (the characteristics are
given, and not separated from the other figures). This is followed by the
principal table, which occupies 11 7 pages. On the first page are given the
logarithmes negatifs of 128 fractions, viz. of all proper fractions whose deno-
minators are CO, 48, 40, or 32, arranged in order thus :— ^-ij, J^, ijV, -gV' sVj
_ _ . .i-I, -^f ,-*o- ^'^^ ^®®*' °^ *^® logarithms are positifs ; and the argu-
ments proce'ed from 1 to 111, with the 128 fractions just described inters
mediate to each integer. Thus we have l^L, l-^, &c., 2-g\,, 2^, &c., as
arguments. The arguments then proceed from 111 to 207 at intervals of
-\, fi-om 2D7 to 327 at intervals of -Jg-, thence to 807 at intervals of |, and
from 808 to 10,400 at intervals of unity, — all to 5 places. The characteristics
are given throughout. A page of proportional parts is added.

There are besides several small tables, to facilitate the calculations, only
one of which requires notice. It gives on a folding sheet the 128 fractions
previously described, expressed as fractions with denominators 100 and 10,
and also (when the numerator is integral) expressed as fractions with de-
nominators 60, 48, 40, 32, 30, 24, 20, 16, 15, 12, 8, 6, 5, 4, 3, 2. Thus ^
= 10A-T-100, and=V4-rlO; as it cannot be expressed in lower terms

58 EEroiiT— 1873.

(or higher terms with any of the above denominators), it only appears as 5 in
the 48 column.

In reference to a work by Girtanner (1794) which we have not seen, but
which appears to be very similar to the present, De Morgan justly remarks,
" But it will not do : Mohammed must go to the mountain. "WheH coin-
age, weights, and measures are decimalized, the use of logarithms will follow
as a matter of course. It is useless trying to bring logarithms to ordinary
fractions."

Rees's Cyclopaedia (Art. " Logarithms," vol. xxi.), 1819. Seven-figuro
logarithms of numbers from 1000 to 10,000, with dififerences ; arranged in
groups of five.

Schron, 1838. Three-figure logarithms to 1400, and five-figure logarithms
to 14,000, with corresponding degrees, minutes, &c., and proportional parts.
Of the 20 pages 4 are occupied with explanations &c. The arrangement is as
in seven-figure tables.

Steinberger, 1840. The titlepage is misleading ; the logarithms do not
extend from 1 to 1,000,000, but only from 1 to 10,000. The only pretext
for giving 1,000,000 as the limit is that, of course, two additional figures may
be obtained by interpolation ; but on this principle ordinary seven-figure
tables should be described as extending, not to 100,000, but to 10,000,000.

The first five figures of the logarithms are printed in larger type than, and
separated by an interval from, the last two, so that the table may be more
conveniently used either as a five- or seven-figure table ; the change of
figure is denoted by an asterisk prefixed to aU the logarithms aflfected. The
figures, though large; are not-clear, the appearance of the page being dazzling ;
the 6's and 9's also seem too large for the other figures, and after all are not
very readily distinguishable from the O's. No differences or proportional
parts are given.

' IVIontferrier's Mathematical Dictionary, 1840. Under the Article
"Logarithines," in t. iii. (the supplementary volume) is given a table of four-
figure logarithms of numbers from 1000 to 10,000 (pp. 271-279).

In the same volume (p. 252, facing letter L) is given a table of logarithms
of the numbers from 1 to 420 to base 2 to five places, the only table of the
kind we have met with.

Babbage, 1841. Seven-figure logarithms of numbers from 1 to 1200 and
from 10,000 to 108,000, with differences and proportional parts (the last
8000 are given to 8 places). Degrees, minutes, and seconds are also added,
but they are divided from the numbers by a thick black line, and are printed
in somewhat smaller type, so that they are not so obtrusive as in Calusi and
others. On the last page there are a few constants.

Great pains were taken with the preparation of this table (which is stereo-
type), with the view of ensuring the maximum of clearness &c., and with
success. The change of figure in the middle of the block is marked by a
change in type in the fourth figure in all the logarithms affected. This is,
we think, with the exception of the asterisk, the best method that has been
used. The chief defect, or rather point capable of improvement, is that the
three leading figures in the logarithms are not separated, or in any way dis-
tinguished, from the rest of the figures in the block, as is the case in Callet
and others. The table was read (wholly or partially) altogether nine times
with different tables of logarithms (four of these readings were made after the
stereotyping), add is no doubt all but perfectly correct.

One feature of this table is that every last figure that has been increased is
marked' with a, dot subscript.

ON MATHEMATICAL TABLES. 59

"We know of oiily two errors : viz., in log 52943 the last figure sKould be
5 instead of 6 ; and in log 102467 the last two figures should be 02 instead of
92, The occurrence of the former of these errors is very remarkable, as the
logarithm is correct in Yega (folio, 1794), with which the table was read
twice (see Sang, ' Athenaeum,' June 8, 1872, and Glaisher, ' AthensDum,'
June 15, 1872, or ' Journal of the Institute of Actuaries,' July 1872 and
January 1873), The latter is given in Gould's (American) ' Astronomical
Journal,' vol. iv. p. 48.

,; Copies of the book were printed on papers of diiferent colours--yeUow,
brown, green, <fec., as it was considered (no doubt justly) that black on a
white ground fatigues the eye more than any other combination *. Tellow
or light brown seem the colours most preferred by computers^ green not being
very satisfactory.

In the preface to his tables (1849), Mr. Fiiipowski writes ; — " Babbage'a-
'Tables of Logaritlims,' which probably are the most accurate of aU; for, by
the aid of his ingenious Qaloulating machine, he was enabled to detect a
variety of errors in former tables." This is untrue.

[Scheutz, 1857.] JFive-figure logarithms, from 1000 to 10,000, calou-'
lated and printed by Scheutz's calculating machine : specimens of a
few other tables are added. A history and description of the machine &c.
is given.

Sang, 1859. Pive-figure logarithms, from 1000 to 10,QOO, arranged as
in a seven-figure table ; no differences.

Gray, 1865. The table in this tract is rather an auxiliary table to
facilitate the calculation of logarithms to twelve places, than a table itself*
The tables at the end of the work (see p. 2 of the Introduction) give
log (l+-001w), log (l+'-OOl'n), log (l + -00r»i), from w=0 to w=999, at
intervals of unity, to twelve places. The use of the sequantities in the cal-
culation of logarithms is' well-known (see, e.g., Introduction to Shoktbede's
Tables, vol. i. 1849). Pages 43-55 are occupied with the history of the
method, and wiU be found valuable and interesting. The rest of the book
is devoted to explanations &c.

"Weddle's method of calculating the logarithms of numbers by resolving
them into the reciprocals of series of factors of the form 1— -l^r, r being a
digit, and then using a subsidiary table of the logarithms of these factors, is
fully explained, as also are some improved methods of Mr. Gray's own,
depending substantially on the same principle ; and aU are illustrated with
fuU numerical examples. The whole constitutes the most complete account
of the simplest and best of the known methods for the calculation of isolated
logarithms that we have met with ; and any one engaged on work of this
kind would do well to consult it. Of course for calculatiug a table, the
method of differences, as Mr. Gray remarks, is the best. A portion of this ■
tract appeared in the ' Mechanics' Magazine ' for 1848 j and the whole is
reprinted from the 'Assurance Magazine and Journal of the Institute of
Actuaries.'

Pineto,. 1871. This work consists of three tables; the first (Table
auxiliaire) contains a series of factors by which the numbers whose logarithms
are required are to be multiplied to bring them within the range of
Table 2, and occupies three pages. It also gives the logarithms of the
reciprocals of the factors to twelve places. Table 1 merely gives logarithms
to 1000, to ten places. Table 2 gives logarithms from 1,000,000 to 1,011,000,

* " Of all the things that are meant to be read, a black monumental inscription on white
marble in a bright light is about the most difficult."— De Morgan.

60 REPOKT— 1873.

to ten places ; the left-hand pages contain the logarithms, and the right-
hand pages the proportional parts, which are given for every hundredth
of the differences. The change in the line is denoted by an asterisk ; and
the last figure is underlined when it has been increased.

The mode of using the tables is as foUows : — If the first figures of the
number lie between 1000 and 1011, the logarithm can be taken out directly
from table 2 ; if not, a factor M is found from the auxiliary table, by which
the number must be multiplied in order to make its initial figures lie between
these hmits, and so bring it within the range of table 2. After performing
this multiplication the logarithm can be taken out ; and to neutralize the

effect of the multiplication, as far as the result is concerned, log f—\ must

be added ; this quantity is therefore given in an adjoining column to M in
the auxiliary table. A similar procedure gives the number answering to any
logarithm, only that another factor (approximately the reciprocal of M) is'
given, so that in both cases multipHcation is used.

The laborious part of the work is the multiplication by the factor M ;
but this is compensated to a great extent by the ease with which, by the
proportional parts, the logarithm is taken out. Greafpains have been taken
to choose the factors M (which are 300 in number) so as to minimize this
labour ; and of the 300 only 25 consist of three figures all different and not
involving or 1. Whenever it was possible, factors containing two figures
alike or containing a 0, or of only one or two figures, have been found. The
process of taking out a logarithm is rather longer than if Viacq or Vega
were used ; but, on the other hand, the size of this book (only about 80 pp.
8vo) is a great advantage, both of the former works being large folios. Also
both Vlacq and Vega are so scarce as to be very difficult to procure ; so that
Pineto's table wiU be often the only ten-figure table available for any one who
has not access to a good library ; and on this account it is unique. Though
the principle of multiplying by a factor, which is subsequently cancelled by
subtracting its logarithm, is frequently employed in the construction of tables,
this is, we believe, the first instance in which it forms part of the process of
nsinff the table. By taking the numbers to 12 instead of 10 places, in a
manner explained in the introduction, greater accuracy in the last place
is ensured than results from the use of Vlacq or Vega. It is not stated
whether the table is stereotyped ; so we presume it is not.

On the last page (p. 56) are given the first hundred multiples of the
modulus and its reciprocal to 10 places. (Notices and examples taken from
Pineto's tables will be found in the ' Quarterly Journal of Mathematics ' for
October 1871, and the ' Messenger of Mathematics ' for July 1872.)

Sang, 1871. Ten-figure logarithms, from 1 to 1000, and seven-figure
logarithms, from 20,000 to 200,000, with differences and multiples (not pro-
portional parts) of the differences throughout.

The advantages arising from the table extending from 20,000 to 200,000,
instead of from 10,000 to 100,000, are, that whereas in the latter the dif-
ferences near the beginning of the table are so numerous that the propor-
tional parts must either be very crowded or some of them omitted, and even
if they are all given the interpolation is inconvenient, in a table extending
from 20,000 to 200,000 the differences are halved in magnitude, while the
number of them in a page is quartered ; the space gained enables multiples
instead of proportional parts to be given.

The table is printed without rules (except one dividing the logarithms
fiom the numbers) ; and the numbers are separated from the logarithms by.

ox MATHEMATICAL TABLES. 61

reversed commas. The absence of rules does not appear to us by any
means an unqualified advantage ; and a further drawback is that numbers
and logarithms are printed in the same type. The change of figure in the
line is denoted by an Arabic nokta (,a sign like the diamond in a pack of cards) ;
and this, though very clear for O's, leaves the other figures unchanged, and
is greatly inferior in all points of view to the simple asterisk prefixed, or the
small figure as used by Babbage.

In spite of these drawbacks the table is very convenient, and has
advantages possessed by no other, as, iii addition to the greater ease with
which the interpolations can be performed, greater accuracy is obtained — the
last figure being often inaccurate by one or two units in logarithms inter-
polated from the usual seven-figure tables. We find, however, that computers
prefer Babbacjb, except for numbers beginning with 1.

The logarithms of the numbers between 100,000 and 200,000 were calcu-
lated de novo by Mr. Sang, as if logarithms had never been computed before ;
and a very fuU account of the method and manner in which the calcula-
tions were performed is given by him in the ' Edinburgh Transactions,'
vol. xxvi. pt. iii. (1871). This is the only calculation of common logarithms of
numbers since the days of Vlacq, 1628 (except the French manuscript tables).
Two errors in the book (which is stereotyped) were pointed out in the
» Athenseum' for June 8 and 15, 1872, viz. the last figures of log 38962 and
52943 should be 2 and 5 instead of 3 and 6 respectively.

Mr. Peter Gray has kindly communicated to us the following six im-
portant errors which have been discovered and communicated to Mr. Sang
(or found on revision) and circulated by him in certain later copies of his
tables ; —

Page 203, log 118536, /or 9503 read 8503 '^
„ „ log 118537, „ 9539 „ 8539*'
„ „ log 118538, „ 9576 „ 8576-^
„ 220, log 127340, „ 9348 „ 9648-^
,. 312, log 173339, „. 9863 „ 8963
,', 354, for number 19540 read 19440,

The following is a classified list of the tables of logarithms contained in
works that are described in § 4 : —

Tables of Logarithms of Numbers (to more than 20 places). — Shaep,
1717 [T. IV.] (61 places) ; Shekwin, 1741 [T. I.] and [T. II.] (61 places)';
HoBBET and Idblek, 17«9 [T. III.] (36 places) ; Byknb, 1849 [T. IV.]
(50 places) ; Callet, 1853 [T. III.], I. and II. (61 places); Httiion, 1858,
T. 5 and 6 (61 places, early editions only) ; Paekhukst, 1871, T. II., III.,
and IX, (102 places),' and T. XVIII. (61 places).

(To 20 places) Gaemneb, 1742, and (Avignon) 1770 [T. IV.] and [T. V.] ;
Pabkhtjesi, 1871, T. XIII. and XIV.

(To 15 places) Douglas, 1809, T. IV., Supplement.

(To 11 places) Boeda and Delambbe, 1800 or 1801 [T. II.] ; Kohlee,
1848 [T. III.]; Caelbx, 1853 [T. II.], I. and II; HoUel, 1858, T. V.
(table to calculate logarithms) ; Htjtton, 1858, T. II. and III.

(To 10 places) De Deceeb, 1626 [T. I.] ; Henkion, 1626 [T. I.] ; ViAca, 1628
and 1631 [T. I.] ; Vlacq, 1633 [T. II.] ; Vega, 1794 [T. I.] ; Hanisohl,
1827, T. IV. ; *SAL0M0ir, 1827, T, VIII. ; Pabkhtjesi, 1871, T. XII.

(To 8 places) John Newton, 1658 [T. I.] ; HotiEL, 1858, T. IV. (table to
calculate logarithms) ; Pabkhubst, 1871, T. XXXVII.
•' (To 7 places) Fatjlhabeb (Logarithmi), 1631 ; Nokwood, 1631 j Eoe, 1633,

62 REPORT— 1873.

T. I. ; ■ OuoirmED, 1657 [T. H.]; Sir J. Moore, 1681 [T. I.].; ViAca,
1681 [T. II.] ; OzANAM, 1685 ; Gaedinek, 1742, and (Avignon) 1770
[T. I.]; Shebwin, 1741 [T. III.]; Dodson, 1747, T. XXXII.; Hentschbn
(VLAca), 1757 [T. II.] ; Sohulze, 1778 [T. I.] j Bonn, 1789, T. I. ; Taylor,
1792 [T. I.] and [T. II.]; Vega, 1797, T. I.; Vega, 1800, T. I.; Borda
and Deiambre, 1800 or 1801 [T. I.] ; .DoxTglab, 1809 [T. I.], and Supple-
ments; Lalande, 1829 [T. I.]; Hassler, 1830 [T. I.]; Gruson, 1832,
T. T.; Turkish Logarithms (1834) ; [De Morgan] 1839 [T. II.] ; Farmy,
1840, T. II. ; HtffissE's Vega, 1840, T. I. ; Troitbr, 1841 [T. IX.] ;
Shoetrede (Tables), 1844, T. I. ; Minsingbr, 1845 [T. I.] ; Kohij;e, 1848
[T. I.] ; Shortredb, 1849, T. I. ; Willich, 1853, T. XX. ; Cailbt, 1853,
T. I.; Bremiker's Vega, 1857, T. I. ; HuiToiir, 1858, T. L; ScheoN, I860*
T. I. ; Wackerbarih, 1867, T. I. ; Dupuis, 1868, T. I. and II.) ; Brtjhns,
1870, T. I.

(To 6 places) DirafN, 1784 [T. I.] ; Adams, 1796 [T. I.] ; Maskbltne (Eg-
quisite Tables, Appendix), 1802, T. III. ; Mackat, 1810, T. XLV. ; Wallace,
1815 [T. I.] ; DucoM, 1820, T. X;XI. ; Lax, 1821, T. XVIII. ; Ebrigajt,
1821, T. X. ; Kiddle, 1824, T. V. ; Ubsinus, 1827 [T. I.] ; Galbraith,
1827, T. II.; *Salomon, 1827, T. VII.; J. Taylor, 1833, T. XVIII.;
NoEDS, 1836, T. XXIV. ; Jahn, 1837, Vol. I. ; Farley, 1840 [T. I.] ; Troxtee,
1841 [T. I.] ; Griffin, 1843, T. 17 ; J. Tayloe, 1843, T. 4 ; EUmker, 1844i
T. I. ; Coleman, 1846, T. XX, ; Kaper, 1846, T. I. ; Domke, 1852, T. XXXII. ;
Beemiker, 1852, T. I. ; Thomson, 1852, T. XXIV. ; Eafee, 1857, T. 64 ;
Beardmoeb, 1862, T. 36; Inman, 1871 [T. Vn.].

(To 5 places) Bams, 1781 [T. I.] ; Maskelyne (Keqnisite Tables), 1802,
T. XVIII. ; BowDiTCH, 1802, T. XVI. ; Lalande, 1805 [T. I.] ; Eios, 1809,
T. XV. ; Moore, 1814, T. IV. ; Db Prasse, 1814 [T. I.] ; PAsauicH, 1817,
T. I. ; Eeynato, 1818 [T. I.] ; Schmidt, 1821 [T. L] ; Stansbury, 1822,
T. X. ; [Schumacher, 1822 ?]. T. V. (arguments in degrees &c.) ; Hantschl,
1827, T. I.; Bagat, 1829, T. XXIII. ; Kohler, 1832 [T. L]; [De Morgan],
1839 [T. I.] ; Gregory &c., 1843, T. XI. ; MtJLLER, 1844 [T. I.] ; Stegmann,
1855, T. I. ; HoUel, 1858, T. I. ; Galbraith and Haughton, 1860 [T. I.],
and [T. II.] ; *ScHt6MiLCH [1865 ?] ; Eankinb, 1866, T. I. ; Wackerbarih,
1867 T I

(To 4 places) [Encke, 1828] [T. I.]; [Sheepshanks 1844] [T. I.];
Warnstorpf's Schumacher, 1845 [T. III.] ; HoUel, 1858, T. VI. ; Anony-
mous [1860 ?] (on a card) ; Oppoizer, 1866.

See also Shoetrede (Comp. Log. Tab.), 1844; Parkhuest, 1871, T.
XXVII. and XXVIII.

Art. 14. Tables of AntihgantJims,

In the- ordinary tables of logarithms the natural numbers are aU integeM,
while the logarithms tabulated are only approximate, most of them being
incommensurable. Thus interpolation is in general necessary in order to
find the number answering to a given logarithm, even to five figures. It
Tvas natural therefore to form a table in which the logarithms were exact
•quantities, -00001, -00002, -00003 to -99999, &c., and the numbers in-
commensurable. Few of such tables have been constructed, as for most
purposes the ordinary tables are sufiiciently convenient, and computers much
prefer to have only one work to refer to. The earliest antilogaiithmic table
is DoDSON, 1742 ; and the only others of any extent are Shortrede (1844
and 1849) arid Filipowski (1849), described in § 4. Mr. Peter Gray has
a large twelve-figure antilogarithmic table far advanced towards completion ;
but whether it will be published is uncertain. - - ■

ON MATHEMATICAL TABLES. 63

Dodsoxi, 1742 (Antilogarithmio Canon). Numbers to eleven places
corresponding to logarithms from '00000 to 1-00000, at intervals of -00001,
arranged like a seven-figure logarithmic table, with intersoript differences,
and proportional parts at the bottom of the page. The changes in the fourth
figure in the middle of the column, bothiu the numbers and the differences^
are marked by points and commas, but not very clearly. There is an intro-
duction of 84 pages ; and the tables occupy about 250 pages.

In page ix of the Introduction an extract is given from Wallis, who states
that Harriot began, and Warner completed, a table of antilogarithms, which
•was ready for press fifty years before. This was told Wallis by Dr. PeU, who
had assisted Warner in the calculation ; and WaUis mentions that he had
himself seen the calculation thirty years before, among Harriot's or Warner's
papers. Dr. Pell subsequently informed Wallis that the papers were in the
hands of Dr. Busby, and that he (Dr. PeU) hoped to pubhsh them shortly.
Dr. Pell died in 1685 ; and at the time Wallis wrote Dr. Busby was
also dead, and the printing had not been begun. Speaking of this manu*-
script De Morgan remarks ; — " AU our efforts to trace it, by help of published
letters ifec, lead to the conclusion that, if existing, it must be among Lord
Macclesfield's unexamined manuscripts at Shirebum Castle : this is by no
means improbable." See, however, some additional information and im-
portant remarks by De Morgan, ' Budget of Paradoxes ' (1872), pp. 457, 458.

A list of thirty-six errors affecting the first eight figures in Dodson's
canon is given by Filipowski in the preface to his ' Antilogarithms ' (1849).
Mr. Peter Gray (' Insurance Eecord,' June 9, 1871) says that in 1847 he had
collected a list of 125 errors in Dodson ; these he communicated to Shoeikedb,
and they were corrected in the plates of his tables (1849). Dodson's work
is unique of its kind, and it remained the only antUogarithmic canon for
more than a century after its completion, tiU. in 1844 Shoeteedb published
the first edition of his tables ; in 1849 he published his second edition ; and
in the same year Pixipowski's tables appeared.

For hyperbolic antilogarithms (viz. e" and e'") see under miscellaneous
tables (§ 3, art. 25).

The following are antilogarithmie. tables described in § 4 :. —

AntUogarithmic Tables. — GARDrNER, 1742, and (Avignon) 1770 [T. VI.]
(20 places) ; Donsoir, 1747, T. XXXIII. ; [Sheepshanks, 1844] [T. YH.].;
Shoeikedb (Comp. Log. Tab.), 1844 ; Shobtrede (tables), 1844, T. II., and
1849, T. II. ; FiLiEowsKi, 1849, T. I. ; Caliet,' 1853 [T. II.], III. ; Stesmanit,
1855, T. IL; Hduel, 1858, T. VL ; Htjtton, 1858, T. IV.; Anontmotts
[1860 ?] (on a card) ; Parkhtjesi, 1871, T. XXVII., XXVHI., and XXXV.

Art. 15. Tables of (Briggian) Logarithmie Trigonometrieal Functions. ■

A general account of the introduction of Briggian logarithms is given in
§ 3, art. 13 ; and NAprEE's ' Canon Mirificus' (1614), containing a Napierian
logarithmic canon, is described in § 3, art. 17. The first table of decimE^
logarithms of numbers was published by Beiggs in 1617, and the first
(decimal) logarithmic canon by Guntee in 1620 (see below), giving the
results to 7 places. The next calculation was by ViAoa, who appended to
his ' Centum Chiliades ' in the ' Arithmetica ' of 1628 a minute logarithmic
canon to 10 places, obtained by calculating the logarithms of the sines &c.
of Rheticits. After the publication of his ' Arithmetica ' in 1624, Beiggs
devoted himself to the calculation 6f logarithmie sines &c., and at his death
in 1631 had all but completed a ten-decimal canon to every hundredth of a

64 REPORT — 1873.

degree. This was published by Vlacq at his own expense at Gouda in
1633, nnder the title • Trigonometria Britannica ' (see below) : the intro-
duction was written by Gellibrand, by whose name the book is sometimes
cited. In the same year Vlacq published his ' Trigonometria Artifieialis,'
containing a ten-second canon to ten decimals. Gunteb's original tablo
contains a good many errors in the last figures ; and a very slight comparison
shows whether any particular table was copied from Gunibk or VrAca?
HENEioif, 1626, and de Decker, 1626 (§ 4), are from the former, Fattlhabee
(§ 4), 1631, from the latter. Briggs appreciated clearly the advantages of
a centesimal division of the quadrant, and, by taking a hundredth of a degree
instead of a minute, made a step towards a reformation in this respect ;
and Button has truly remarked that, but for the appearance of Vlacq's
work, the decimal division of the degree might have become recognized,
as is the case with the corresponding division of the second*.

The next great advance on the ' Artifieialis ' was more than a century and
a half afterwards, when Michael Taylor (1792) published bis seven-decimal
canon to every second (§ 4). On account of its great size, and for other reasons,
it never came into very general use, Bagat's 1829 (§ 4) being preferred ;
the latter is now, however, very difficult to procure. The only other canon
to every second we have seen or heard of is Shorteede's, 1844 and 1849
(§ 4), which is the most complete as regards proportional parts &c. that we
know of. The canon is in modern editions issued separately.

Lalande (' Encyclopedic Methodique. Math^matiques,' Ast. Tables) states
that in AprU. 1784 he received from M. Eobert, cure of St. Genevieve at
Toul, a volume of sines for every second of the quadrant, and soon after
the tangents; but he had heard that Taylor, in England, was engaged in
publishing log sines and cosines to every second, and that the Board of
liongitude had contributed £300 to the expense. These volumes were pur-
chased by Babbage at the sale of Delambre's library, and they appear in the
Babbage Catalogue (only the title of the table of sines is given ; but it is to
be presumed that the library contains both, as two volumes arc spoken of).
Babbage lent them in 1828 to the Board of Longitude ; and some errata in
Taylor, 1792, were found by means of them. [They are now (1873) in the
possession of Lord Lindsay, who has purchased the whole of Mr. Babbage's
mathematical library.]

No ten-decimal canon to every second has been calculated. The French
manuscript tables are described in § 3, art. 13. Of logarithmic trigonometrical
canons that have appeared the number is very great. We may especially
jnention Callei, 1853; Beemikeb's Veoa, 1857; HiriioN, 1858; Scheon,
1860 ; Dupuis, 1868; and Betjhns, 1870.

* The centesimal division of the degree is of paramount importance, whereas the cente-
simal division of the right angle is of next to none at all ; and had the French mathemati-
cians at the end of the last centuiy been content with the former, it is not unlikely that their
tables would have superseded the sexagesimal ones still in use, instead of having been almost
totally ignored by computers. The hundredth part of a right angle is almost as arbitrary a
unit as the ninetieth ; and no advantage (but on the contrary great inconvenience) would rt-
eultfrom the change ; but to divide the nonagesimaldegree into centesimal minutes, and these
into centesimal seconds, &c., in other words to measure angles by degrees and decimals of
a degree, would ensure all the advantages of a decimal system (a saving of work in interpo-
lations, multipUcations, &c.). This Briggs and his followers, Eoe, Oughtred, John Newton,
&n., perceived and acted upon two hundred and fifty years ago ; and they seem to have
shown a truer appreciation of the matter than did the French mathematicians. It may
be taken for granted that the magnitude of the degree will never be altered ; but there is
no reason why sexagesimal minutes and seconds should not be replaced by decimals of a,
degree; and this is a change which might, and we hope will heresrfter be made.

ON MATHEMATICAL tABLES. 65

The chief tables in which the angle is divided completely centesimally are
Callet 1853, BoKDA and Delambre, and Hobeei and Idelek.

Porthe meaning of S and T (Dclambro's tables), see § 3, art. 13, near the
end of the introductory remarks.

Gunter, 1620. Log sines and tangents for every minute of the quadrant
(semiquadrantally arranged) to 7 places; This is the first (Briggian) loga^
lithmic trigonometrical canon calculated or published. The book is ex-
tremely scarce ; and we have only seen one copy of it, viz. that in the British
Museum, where it is bound up with Bkiggs's ' Logarithmorum Chilias Prima.'
There is engraved on the titlepage a diagram of a spherical triangle, S P Z,
De'Morgan (who had never seen a copy) says -that it also contains logarithms
of numbers as far as 1000 _; but this is not correct. The British-Museum copy
has written in ink on the titlepage, " Eadius autem verus est 10,000,000,000."
This has reference to the fact that the logarithm of the radius is taken
to be 10, and is true in one sense, but not in the usual one, which
is that, this being the radius, the sines &c. are true to the nearest unit.
Custom has very properly decided to consider the radius of a logarithmic
canon the same as what would be the radius of the resulting natural canon
if the logarithms were replaced by their numbers. "We have not seen the
second edition, in which, no doubt the logarithms of numbers mentioned
by De Morgan were added ; or it is just possible that some copies of
Briggs's ' Chilias ' (1617) were issued with the ' Canon,' both being bound
together in the copy we have seen, and that this has given rise to the
assertion. Gfnter's ' Canon ' was also issued under an English title, ' A
Canon of Triangles,' &c. (Bodleian Catalogue) : see Phil. Mag. (Suppl. No.)
Dec. 1872. Por a life of Gunter, see Ward's ' Lives of the Professors of
Gresham College,' pp. 71-81.

. Briggs, 1633 (' Trigonometria Britanuica '). .Natural sines (to 15
places) and tangents and secants (to ' 10 places), also log sines (to 14
places) and tangents (to lO places), at intervals of a hundredth of a degree
from 0° to 45°,- with interscript differences, for all the functions. The
division of the degree is thus centesimal; but the corresponding argu-
ments in minutes and seconds are also given, the intervals so expressed
being 36".

This table was calculated by Briggs ; but he did not live to publish it. The
trigonometry is by Gellibrand.

Gunter, 1673. At theend of the work is given a table of log sines and
tangents for every minute of the quadrant to 7 places, followed by seven-
figure logarithms of numbers to 10,000.

The table of log sines &c. is printed as it appeared in Gttniee's ' Canon
Triangulorum,' 1620, as the last figures in very many instances differ from
the correct values, which were first given by Ylacq in the ' Arithmetica ' &c.
(1628).

This is the fifth edition of Gunter's works ; but we remember to have seen
it stated somewhere that the works themselves (separate) were regarded
as the first edition in this enumeration.

Berthoud, 1775. At the end of the ' Reoneil des Tables n^cessaires
pour trouver la longitude en mer,' is a table of log sines to every minute of
the quadrant to 6 places (pp. 25-34).

Callet, 1827 (' Log Sines &c.'). Log sines and tangents for every second
to 5°, and log sines, cosines, tangents, and cotangents from 0° to 45°, at
intervals of ten seconds, with differences, all to seven places.

1873. !■

66 KEPORT— 1873.

These are the same as Callet 1853 [T. IX. and X.] (§4), and were pub;
lished separately, De Morgan states, to accompany Babbage's logarithms of
numbers ; they are in consequence printed on yellow paper ; but it is, both
in colour and texture, very inferior to that used by Babbage.

Airy, 1838. Log sines and cosines from 0" to 24", at intervals of
10' to 5 places. The proper sign is prefixed to each quantity : no differ-
ences. The sines are on the left.^hand pages, the cosines on the right-hand.
As was remarked by De Morgan, this is an eightfold repetition of one
table : it occupies 48 pp. The table is improperly described as having been
" computed under the direction " &c. : it is, of course, only a simple re-
arrangement.

The following is a complete classified list of tables on the subject of
this article contained in the works that are described in § 4, with several
other lists appended.

Loff sines, tangents, semnts, and versed sines. — (To 7 places) Willich,
1853, T. B ; Button, 1858, T. IX.

(To 5 places) Eios, 1809, T. XVI. (also log coversed &e.).

Log sines, tangents, and secants. — (To 10 places) Viacq, 1628 and 1631
[T. II.]; Faitlhaber (Canon), 1631.

(To 7 places) Sir J. Mooee, 1681 [T. III.] ; Shekwin, 1741 [T. IV.] ;
BoKDA and Delambke, 1800 or 1801, T. VI. (centesimal) ; Dotiglas, 1809
[T. II.].

(To 6 places) Dunn, 1784 [T. II.] ; Adams, 1796 [T. II.] ; Wailacb,
1815 [T. n.] ; J. Tatloe, 1833, T. XIX. ; Nome, 1836, T. XXV. ; Teotteb,
1841 [T. in.]; Geiitin, 1843, T. 18 ; J. Tatior, 1843, T. 5; RtJMKER,
1844, T. n. ; Coleman, 1846, T. XXIII. ; Raper, 1846, T. IV. ; Domke;
1852, T. XXXV. ; Eapee, 1857, T. 68; Inman, 1871 [T. IV.].

(To 5 places) Maskelyne (Requisite Tables), 1802, T. XIX.; Bow-
DITCH, 1802, T. XVII.; Moore, 1814, T. V.'; Galbraith, 1827, T. V,;
Gbegoet Ac, 1843, T. IX. ; HoUel, 1858, T. II.

(To 4 places) Gordon, 1849, T. IX. (cosecants).

Log sines and tangents (only). — (To 11 places) Boeda and Delambee, 1800
or 1801 [T. III.] (centesimal), and [T. V.] (logarithmic dilferences of sines
and tangents).

(To 10 places) Vlacq, 1033 [T. I.]; Roe, 1633, T. I. (centesimal
division of the degree) ; Vega, 1794, T. II.

(To 8 places) John Newton, 1658 [T. II.] and [T. IIL] (arguments
partly centesimal).

(To 7 places) de Decker, 1626 [T. II.] ; Heneion, 1626 [T. II.] ; NoRwooDi
1631 ; ViAca, 1681 [T. I.] ; Ozanam, 1685 ; Gardiner, 1742, and (Avignon),
1770 [T. II.]; DoDsoN, 1747, T. XXXIV.; Henischen (Vlacs), 1757
[T. I.]; ScHULZE, 1778 [T. IH.] and [T. V.]; Donn, 1789, T. IIL;
Tatlor, 1792 [T. IIL] ; Vega, 1797, T. U. ; Lambert, 1798, T. XXVI. ;
HoBERT and Idelisb, 1799 [T. I.] (centesimal) ; Vega, 1800, T. II. ; (?) «Salo-
MON, 1827, T. IX.; Bagay, 1829, Appendix; Lalande, 1829 [T. II.];
Hassler, 1830 [T. H.-IV.]; Gbuson, 1832, T. VII. ; Turkish logarithms
[1834] ; HtossE's Vega, 1840, T. II. ; Shoetrede (Tables), 1844, T. IIL,
and 1849, Vol. II. ; Kohler, 1848 [T. IV.] ; Callet, 1853 [T. VI.] (cente-
simal), [T. IX.] and [T. X.] ; Beemikeb's Vega, 1857, T. II. and HI. ;
Huiton, 1858, T. VIII. ; Scheon, 1860, T. II. ; Dupuis, 1868, T. VI., VII.,
and VIII. ; Bruhns, 1870, T. H. and III.

(To 6 places) Oughtred, 1657 [T. I.] (centesimal division of degree) ;
DucoM, 1820, T. IX. ; TJrsinus, 1 827 [T. TL] and [T. V.] ; J. Taylor, 1833,-

ON MATHEMATICAL TABLES. 67

T. XIX.; Nome, 1836, T. XXV.; Jahw, 1837, Vol. II.; FAELEr, 1840
[T. II.] ; J. Taylob, 1843, T. 5 ; Eumkee, 1844 ; Domke, 1852, T. XXXIV. ;
Beemikeb, 1852, T. II.

(To 5 places) Bates, 1781 [T. II.] ; Lalanbe, 1805, T. II. ; De Pkasse,
1814 [T. II.] ; PAsaxjicH, 1817, T. II. ; EExifAra, 1818 [T. II.] ; Schmidt,
1821 [T. II.]; Koheeb, 1832 [T. II.]; [Db Morgan], 1839 [T. III.];
(Jalbeaixs and HATOHioif, 1860 [T. III.] ; Waokeebartjec, 1867, T. III.

(To 4 places) [Eitoke, 1828] T. II. ; Bevbkeet (1833 ?), T. XVII. ;
MtJxtEE, 1844 {T. IV.]; [Sheepshaitks, 1844J [T. HI.]; Waeustoeef's
ScHTOACHEE, 1845 [T. IV.] ; Thomson, 1852, T. XVI. ; Oppolzbe, 1866 ■
Paekhuest, 1871, T. XXX. and XXXI.

(Miseell.) Shorieede (Gomp. Log. Tab.) 1844.
Log sines and secants (only).— {To 5 places) Stansbttet, 1822, T. E.
Log sines (alone*) (for small arcs, sines = tangents), — (To 7 places)
Gaedinee, 1742 [T. II.], and (Avignon) 1770 [T. II.] ; Hitmsb's Vboa, 1840,
T. II. ; KoHLBE, 1848 [T. IV.].

(To 6 places) Mackat, 1810, T. XLVI. ; Keriqan, 1821, T, VIII. s

Hajtiscbl, 1827, T. II. ; FAELEr, 1840 [T. III.] ; Eapee, 1846, T. III. ;

Eaper, 1857, T. 66 and 67 ; Beardmoee, 1862, T. 37 ; Inman, 1871 [T. III.],

(To 5 places) [Schumachee, 1822?] T. VI.; [Db Moeoan] 1839 [T. IV.] ;

Eapee, 1846, T. II. ; Thomson, 1852, T. XII.

(To 4 places) [Sheepshanks, 1844] [T.II.]; Paekhtjest, 1871, T. XXXVIII.
(Expressed otherwise) Acai)4mie be Peussb, 1776 [T. I.] ; Caelbt, 1853
[T. VII.] (centesimal) (15 places).

Log tangents (alone*) (for small arcs, sines = tangents).- — (To 7 places)
Gaedinee (Avignon), 1770 [T. II.].

(To 6 places) Mackat 1810, T. XLVII. ; Hantschl, 1827> T. III.
Log versed sines (alone). — (To 7 places) Sir J". Mooeb, 1681 [T. IV.] ;
[Sir J. MooEE, 1681, versed sines] ; Douglas, 1809 [T. IV.] ; Faelet, 1856
[T. II.].

(To 6 places) Eumker, 1844, T. IV.

(To 5 places) Keeigan, 1821, T. XI.; J. Taylok, 1833, T. XXL, and
1843, T. 30.

(To 4 places) Bonn, 1789, T. V»

Note. — Log rising (in nautical tables) = log versed sine. See next page.
Log secants (alone). — (To 5 places) Thomson, 1852, T. XI.
Miscellaneous. —Log sec a;, -J- log sec x, and | log sin x, Ceoswell, 1791j
T. I. ; log diff. sin., Boeda and Delambee, 1800 or 1801 [T. V.] (centesimal) ;
log I (1 + cos x), log i (I ± sin x) &e., Eios, 1809, T. XVI. ; log tan

'-, SiANSBUKY, 1822, T. 23; logi (1 - cos x) &c., Stansdtjet, 1822, T. B. -,

log 1 (1 —cos x), NoEiE, 1836, T. XXXI. ; log ^ (1-cos x), Coleman, 1846,
T. XXL; log I (1— cos x), Goedon, 1849, T. XVIII. ; log 1 (1-cos x),
Thomson, 1852, T. XIII. ; log coscc x — 54000, Thomson, 1852", T. XV. ; log

sin f, Thomson, 1852, T. XXIIL; log ^ (1-eos x), Eaper, 1857, T. 69;

4- log J (1 — cos x) and log| (1— cos x), Inman, 1871, T. V. and VI.
The following are tables generally met with in nautical collections :—
Log sines, tangents, and secants to every quarter-point. — (To 7 places)

* Tables of sines and tangents are not unfrequently printed willi the sines on the versos
and the tangents on the rectos of the leaves, or vice versa, so that practically they are sepa-
rated ; but in such cases the table has usually been regarded merely as one of sines and
tangents.

t 2

G8 REPORT— 1873.

NoRiE, 1836, T. XXIII. ; Shortrede (Tables), 1844, T. V. ; Donn, 1789,
T. II. (sines and cosecants only).

(To 6 places) Eiddle, 1824, T. IV. ; Gaibraith, 1827, T. IV. ; J. Taylor,
1833, T. XVII. ; Trotter, 1841 [T. II.] ; GRiFFiif, 1843, T. 16 ; J. Taylor,
1843, T. 3; CoLEiiAir, 1846, T. XIX.; Domke, 1852, T. XXXII.; Raper,

1857, T. 65.

(To 5 places) Adams, 1796 [T. III.]; Bowditch, 1802, T.XVL; Moore,
1814, T. III.

Log. I elapsed lime, mid time, and nsing. — (To 5 places) ,DoNjr, 1789,
T. IV. ; Maskelyne (Requisite Tables), 1802, T. XVI. ; Bowditch, 1802,
T. XIII.

The three Tables are separated in the following: — (To 5 places) Mackay,
T. XLVIII.-L. ; Moore, 1814, T. XXIII. ; Norie, 1836, T. XXVII.-
XXIX. ; DoMKB, 1852, T. XXXVII.-XXXIX.

We have thought it worth while to collect into one list below all the tables,
giving log sines &c. to every second. It must be particularl}- noticed, how-
ever, that in the great majority of cases only the functions for the first few
degrees of the quadrant are given to every second in the tables referred to,
which should in all cases be sought in § 4.

Tables of logarithmic trigonometrical functions to seconds. — Gardiner,
1742 [T. II.I, and (Avignon) 1770 [T. II.]; Schulze, 1778 [T. III.];
Taylor, 1792, T. III. (for the whole quadrant) ; Veka, 1794, T. ll. ; Vega,
1797, T. II. ; Vega, 1800, T. II. ; Dtcom, 1820, T. IX. ; Xbrigan, 1821,
T. VIII. ; [Schumacher, 1822 ?] T. VI. ; *SAL0M0sr, 1827, T. IX. ; Baoay,
1829, Appendix (for the whole quadrant) ; Hassleh, 1830 [T. II.] ; Jahn,
1837, Vol. II. ; [De Morgan] 1839 [T. IV.] ; HtJLssE's Vega, 1840, T. II. ;
MCtLER, 1844 [T. IV.] ; Shortredb (Tables), 1844, T. III. and 1849,
Vol. II. (for the whole quadrant); Eaper, 1846, T. II.; Kohlbr, 1848
[T. IV.] ; Domke, 1852, T. XXXIV. ; Bremikee, 1852, T. II. ; Callet, 1853
[T. IX.] ; Bremiker's Vega, 1857, T. II. ; Raper, 1857, T. 66 ; Hution,

1858, T. VIII. ; Wackbubarth, 1867, T. III. ; Dupuis, 1868, T. VI. and
VII. ; Brums, 1870, T. II. ; Inman, 1871 [T. III.] and [T. VIII].

We have formed the following lists of tables in § 4, which (not only in thd
same work, but side by side in the same table) give both natural and
logarithmic functions : —

Tables containing both natural and logarithmic functtom (in the sami table).
—(To 15 places) Callet, 1853 [T. VIL] (centesimal).

(To 7 places) Sir J. Moore, 1681 [T. III.]; Vlaco, 1681 [T. I.] ;'
OzANAM, 1685 ; SnERWiN, 1741 [T. IV.] and [T. V.] ; Huntschen (Vlacci),
1757 [T. I.] ; Schulze, 1778 [T. V.] ; Bonn, 1789, T. III. ; LAMnEEi, 1798,
T. XXVI. ; Hobekt and Ideler, 1799 [T. I.] (centesimal) ; Willich, 1853,
T. B ; HuTioN, 1858, T. IX.

(To 6 places) Oughtred, 1657 [T. I,]; ITksinus, 1827 [T. V.].

(To 5 places) Hotjel, 1858, T. II.

(To 4 places) Donk, 1789, T. V.

(Mixed) Bates, 1781 [T. II.].

Natural and log versed sines (in the same table). — (To 7 places) Sir J. Moore,
1681 [T. IV.] ; [Sir J. Moore, 1681, versed sines]; Sheewin, 1741 [T. V.l •
Pouglas, 1809, T. IV.

Art. 16. Tables of Hyperbolic Logarithms (viz. logarithms to base 2'71828. . .),

The logarithms invented by Napier, and explained in the ' Descriptio '

(1614) and ' Constructio ' (1619) (see § 3, art. 17), were not the same as

ON MATHEMATICAL TABLES. 69

those now called hyperbolic (viz. to base e) and very frequently also I^ajnerian
logarithms. It is also to be noticed that Napier calculated no logarithms of
numbers. Jonx SrEiDELL, 1619 (see below), first published logarithms to
base e both of numbers and sines. The most complete table of hj'perbolio
logarithms is Base's, described below, which could be used, thougla not so
conveniently, as an ordinary seven-figure Briggian table estending from 1000
to 105,000. It would sometimes be useful to have also a complete seven-
place table of hyperbolic logarithms of numbers from 1000 to 100,000, ex-
actly similar to the corresponding' Briggian tables, as in some cases it is con-
venient to perform calculations in duplicate, first by Briggian, and then by
hyperbolic logarithms ; and such a table would be of use in multiplying five
figures by five figures : but hyperbolic logarithms cannot be rendered conve-
nient for general purposes.

The most elaborate hyperbolic logarithmic table is Wolfram's, which prac-
tically gives the hyperbolic logarithms of all numbers from unity to 10,000
to forty-ciglit decimal places. It first appeared, we believe, in Schtoze (§ 4),
and was reprinted in VbctA, folio, 1794 (§ 4).

"Wolfram was a Dutch lieutenant of artillery ; and his table represents six
j-ears of very laborious work. Just before its completion he was attacked by
a serious illness ; and a few logarithms were in consequence omitted in Schtoze
(see Introduction, last page but two, to vol. i. of Schtoze). The omissions
■were supplied in Vega's ' Thesaurus,' 1794. De Morgan speaks of Wolfram's
table as one of the most striking additions that have been made in the sub-
ject of loganthms in modern times.

Montucla (' Histoire,' vol. iii. p. 360) states that in 1781 Alexander Jom-
bert proposed to publish by subscription new tables of hyperbolic logarithms
to 21 places for all prime numbers to 100,000, with a table of all odd numbers
of two factors to the same limit. The author was Dom Vallcyre, advised by
Dom Robe, benedictine of St. Maur. Only two hundred subscribers were re-
quired before the commencement of the printing, and nothing -was asked in
advance; but the project fell through, no doubt for want of subscribers.
We infer from this account that the table vcas calculated.

The Catalogue of the Eoyal Society's Library contains, under the name of
Prony, the title, " Eormules pour calculer I'effet d'une machine a vapeur a

detente et a uu seul cylindre Tables de logarithmes hyperboliques calcu-

lees de 100° en 100° d^unite, fol. lithog.," but without any reference to the
place where the book is to be found in the library, so that -we have not seen it.

Speidell, 1619. Logarithmic sines, tangents, and secants, semiquadrantally
arranged, to every minute, to five places. The logarithms are hyperbolic (viz.
to base e), and the first of the kind ever pubhshed. When the characteristic
is negative SpeideU adds 10 to it, and does not separate the characteristic so
increased from the rest of the figures by any space or mark. Thus he prints
the logarithm of the sine of 21° 30' as 899625, itstrue value being 2-99625 ;
but the logarithm of the cotangent is given as 93163 ; it would now bo
written -93163. The Eoyal Society has " the 5-impression, 1623," with the
" Breefe Treatise of Sijhffiricall Triangles " prefixed, and also some ordinary
hyperbolic logarithms of numbers (the first published) &c. On this see De
Morgan's long account of gpeidell's works, who, however, had never seen the
edition of 1619, in which the canon occurs by itself without the logarithms
of numbers. We cannot enter into the question of Speidell's fairness here.
The 1619 copy we have seen (Cambridge Univ. Lib.) has an obliteration
where, in the 1623 copy, there occur, the words " the 5-impression."

70 KEPOKT-.1873.

Rees*s Cyclopaedia, 1819 (Art. " Hyperbolic Logarithms," vol. xviii.).
Hyperbolic logarithms (to 8 places) of all numbers from 1 to 10,000, arranged
in groups of five.

The table was calculated by Barlow, and appears also in. his mathema-
tical tables (1814).

Dase, 1850 (Hyperbolic Logarithms). Hyperbolic logarithms, from
1 to 1000, at intervals of unity, and from 1000-0 to 10500-0 at intervals
of 0-1 to seven, places, with differences and proportional parts, arranged
as in an ordinary seven-figure table. The change of figure in the line is do-
noted by an asterisk prefixed to aU the numbers affected. The table is a
complete seven-place table, as by adding log 10 to the results the range
is from 10,000 to 105,000 at intervals of unity. The table appeared in the
34th part (new series, t. xiv.) of the ' Annals of the Vienna Observatory'
(1851); but separate copies were printed, in the preface to which Base gave
six errata. This portion of the preface is reproduced in the introduction by
Littrow to the above volume of 'Annals.' The table was calculated to ten
places, and three rejected. It was the author of this table who also com-
puted the factorial tables (§ 3, art. 8), and calculated the value of tt cor-
rectly to 200 decimal places (CreUe's Journal, t. xxvii. p. 198).

Filipowski, 1857. Hyperbolic logarithms, from 1 to 1201, to 7 places,
are appended to Mr. FUipowski's English edition of Napier's ' Canon
Mirificus.'

The following is a list of references to § 4 : —

Eyperholie logarithms of numbers. — (To 48 places) Schulze, 1778 [T. II.] ;
VESA, 1794 [T. III.l ; Cailet, 1853 [T. III.], I., and II.

(To 25 places) Lambert, 1798, T. XVI.

(To 20 places) Callet, 1853 [T. H.], I. and 11.

(To 11 places) Borda and Delambre, 1800 or 1801 [T. IV.].

(To 10 places) »Salomon, 1827, T. VIII.

(To 8 places) Ve&a, 1797, Vol. II. T. II. ; Barlow, 1814, T. VI. ; Hant-
BCHL, 1827, T. VI. ; HtLssE's Vega, 1S40, T. VI. ; Trotter, 1841 [T. XI.] ;
KoHLEE, 1848, T. I.

(To 7 places) Gardiner (Avignon), 1770 [T. VIL] ; Lambert, 1798,
T. XIII.-XVI. ; WiLLicH, 1853, T. A ; Htttton, 1858, T. V. and VI. :
DupiTis, 1868, T. III.

(To 5 places) Rankinb, 1866, T. 3 ; "Wackeebarth, 1867, T. V.
, See also *ScHL0MiLcn [1865 ?].

Art. 17. Napierian Logarithms {not to base 2-71828. . . .).
The invention of logarithms has been accorded to Wapier of Merchiston
with a unanimity not often met with in reference to scientific discoveries.
The only possible rival is Justus Byrgius, who seems to have constructed a
rude kind of logarithmic table ; but there is every reason to believe that
Napier's system was conceived and perfected before Byrge's in point-of time -
and in date of publication Napier has the advantage by six years. Purthcr
B3Tge's system is greatly inferior to Napier's ; and to the latter alone is the
whole world indebted for the knowledge of logarithms, as (with the exception
of Kepler, one of the most enthusiastic of the contemporary admirers of
Napier and his system, who does allude to Byrge) no one ever sugwested
any one else as having been the author whence they had drawn their
information, or as having anticipated Napier at all, till the end of the last
century, when Byrge's claim was first raised, though his warmest advocates
always assigned far the greater part of the credit of the invention to Napier.

ON MATHEMATICAli TABLES. 71

On Byrge's claim see De Morgan's careful resume (article " Tables," under
Justus Bypgius, 1620, in the 'Eng. Cyclop.,' where references are given),
and Mr. Mark Napier's ' Memoirs of John Napier of Merohiston,' Edin-
burgh, 1834 (where the question how far Napier received any assistance
Jrom his predecessors in the discovery is fully discussed). We have also seen
' Justus Byrg als Mathomatiker und dessen Einleitung in seine Logarith-
men,' by Dr. Gieswald, Dantzig, 1856, 4to (pp. 36). Napier's ' Canonis
Logarithmorum Miriiici Descriptio ' (which contained the first announcement
and the first table of logarithms) was published in 1614 ; and in 1619 (two
years after his death, which occurred on April 4, 1617) appeared the ' Mirifioi
logarithmorum Canonis Constructio,' edited by his son Kobert, in which the
method of constructing the canon is explained. The various reprints and
translations of the ' Descriptio ' and ' Construetio ' are described under
Napieii, 1614 and 1619 j and the relations between Napiee and Bhigos with
regard to the invention of decimal logarithms are noticed in § 3, art. 13.
The most elaborate canon of Napierian logarithms is UBsnrus (1624-1625),
described below.

The difference between the logarithms introduced Napier and hyperbolic
logarithms is explained under Napibe (1614). "We have paid considerable
attention to the early logarithmic tables, and have examined all of them that
were accessible to us ; and it is with some regret that we omit to notice them
in detail here : the accounts of the smaller tables that immediately suc-
ceeded Napier would be of only bibliographical or historical interest ; and to
describe them with sufiicient detail to render the accounts of value would
occupy too much space. However, as the works of this period are very rare,
it is worth while remarking that there is a copy of Napier's ' Constructio '
in the Cambridge University Library (there is none in the British Museum
or Koyal Society's Library), where also are to be found Ursinus's ' Cursus ' of
1618, Speidelt. 1619, and Keplee 1624 : we have generally, in describing
works of this date, mentioned the library containing the copy we have seen.
We have found De Morgan to be very accurate (except where he has had to
form his opinions from secondhand or imperfect evidence) ; and he has
■devoted much care to the early logarithmic tables, so that we feel the less
reluctance in omitting to notice them further here.

INTapier, 1614. The book consists of 57 pp. explaining the nature of
logarithms &c., and 90 pp. of tabular matter, giving natural sines and their
Ncipierian logarithms to every minute of the quadrant (semiquadran tally
arranged) to seven or eight figures (seven decimals). Logarithmic tangents
.are also given under the heading (^tjfer«n«ice (they are the differences between
the sine and cosine, which, though the latter name is not used, are both on
the same line, as a consequence of the semiquadrantal arrangement of the

table). ^^ . .

The logarithms introduced by Napier were not hyperbohc or JNapienan
logarithms as we now understand these terms, viz. logarithms to the basee
(2-71828 . . ), but somewhat different ; the relation between the two being

i = lO'' e~io^ or L = 10' log, IC - 10' I,
I being the logarithm to base e, and L the Napierian logarithm ; the relation
between N (a sine) and L, its Napierian logarithm is therefore

N = 10,000,000 e 10,000,0005

72 REPoiiT— 1873.

the logarithms therefore decrease as the sines increase. A brief exphmation
of the principle of Napier's own method is given by Professor "Wackerbarth
in vol. xxxi. p. 263 (1871) of the 'Monthl}- Notices of the lloyal Astro-
nomical Society.' The author of that communication there points out that
the description in most elementary books of Napierian logarithms, as loga-
rithms to the base e, is incorrect; but this criticism appears to us irrelevant,
as by Ciilhng certain logarithms Napierian it is not asserted that they are
used at present in the exact form in which they were presented by Napier.
A glance at the formula written above shows that aU the essential features
of logarithms to the base e are contained in Napier's system, and that there
is no impropriety in calling the former by his name. De Motgan says that
" Delambre proposed to call them [Napier's logarithms] Napierian logarithms,
and to restrict the term hyperbolic to the modern or e logarithms ; but
custom has refused," — and no doubt very properly, as, except in mathematical
histories &c., there is no occasion to distinguish the two systems from one
another. For our own part, we should much prefer to see natural or
hyperbolic and common logarithms universally called Napierian and Briggian,
after the two great founders of logarithmic tables.

A translation of Napier's ' Canon Miriflcus ' was made by Edward Wright
(well known in connexion with the history of navigation), and, after his death,
published by his son at London in 1616, under the title " A Description of
the admirable Table of Logarithmes, &c." (12mo). On account of the rarity
of this wort and the ' Constructio,' the fuU titles of both are given in § 5.
There is a short " Preface to the Eeader " by Briggs, and a description of a
triangular diagram invented by Wright for finding the proportional parts.
Napier's table, however, is printed to one figure less than in the ' Canon
Mirificus' throughout. The edition was revised by Napier himself. On
Wright, see Introduction to Hutton's ' Mathematical Tables.' The ' Canon
Mirificus ' was also reprinted by Maseres in the sixth volume of the ' Scrip-
tores Logan thmici' (1791-1807); and in 1867 Mr. PiLirowsKi published
at Edinburgh a translation of the same work (full title given in § 5 ; the tone
of the Introduction renders any comment on it unnecessary).

Both the ' Descriptio ' (the ' Canon Mirificus ') and the ' Constructio '
wore reprinted by Bartholomew Vincent at Lj'ons in 1620 (who thus first
published logarithms on the Continent), the title of the former appearing on
the titlepage as " Logarithmorum Canonis Descriptio, sen Arithmeticarum
supputationum mirabilis abbreviatio. Ejusque usus in utraque Trigonometria
nt etiam in omni Logistica Mathematica, amplissimi, faoillimi & expeditissimi
oxplicatio. Authore ac Inventore Joanne Nepero, Baroue Merchistonii, &c.,
Scoto. [Printer's device with word Vincenti.l^ Lugduni. Apud Barth. Yin-
contium, M.DC.XX. Cum priyilegio Cassar. Majest. & Christ. Galliarum
Eegis." The full title of Napier's original edition of 1614 is given in § 5 ;
and it wiU be seen that it is very difierent from that written above. Yery
many writers (including Montucla) give the title of Yincent's reprint as that
of the original work. There is an imperfect copy of Yincent's reprint,
containing only the ' Descriptio ' (the ' Constructio ' having been torn out),
in the British Museum ; but the Eoyal Society has a perfect copy. Wright's
translation of 1616 is in the British Museum.

On the accuracy of Napier's Canon see Delambre, < Astron. Mod.,' t. i.
p. 501. Mr. Mark Napier's ' Memoirs of John Napier ' gives nearly all that
is known with regard to Napier's life, M8S., &ei;;biit it is told in a verbose
and diffuse manner, and written in a partisan spiritas regards Briggs.

A manuscript on arithmetic and algebra, writ;ten by Napier and left by

ON MATHEMATICAL TA11L33S. 73

Mm to Briggs, was privately printed in 1839, under the title " De Arte
Jjogistica Joannis Naperi Merchistonii Baronis libri qui supersuut," edited by-
Mr. Mark Napier. An historical sketch, mainly derived from the same
author's ' Memoirs,' is prefixed. In 1787 was also published ' An account
of the Life, Writings, and Inventions of John Napier of Merchiston,' by
David Stewart, Earl of Buchan, and Walter Minto, LL.D. Perth, 4to. See
also Phil. Mag. Suppl. No., December, 1872, " On some early Logarithmic
Tables." Leslie ('Philosophy of Arithmetic,' 2nd edit., 1820, p. 246)
describes Napier's work as " a very small duodecimo ;" the last word should
be " quarto." The page is 7-7 inches by 6-7 inches.

We may remark that Napier's name is spelt in a variety of ways ; we
have seen Neper, Naper, Nepair, and Nopper. He always Latinized his
name into Neperus or Naperus, but spelt it in the vernacular several ways.
The family now write the name Napier; and this spelling is generally
adopted, and with good reason.

N'apier, 1619 (' Construotio '). This work containsno table, and is there;-
fore not properly included in this Eeport. We have, however, noticed it on
account of its being a sequel to the ' Descriptio,' and also on account of its
rarity (the full title is given in § 5). The only copy we have seen (in the
Cambridge University Library), which belonged to Oughtred, contains two
titlepages, the first running "Mirifici logarithmorum canonis descriptio. . . .
aceesserunt opera posthuma ; prime, Mirifici ipsius canonis construotio . . . ^
Edinburgi. . . .1619," and the second being as given in § 5. Prom this we
infer that a reprint of the 'Descriptio' (1619) was prefixed to the
' Constrnctio,' but that it was torn out from the copy we have examined.

On the reprints, &c. of the ' Construotio,' see under Napiek, 1614.

Ursiuus, 1624-1625. A canon exactly similar to Napier's in the
' Canon Mirificus,' 1614, only much enlarged. The intervals of the argu-
ments are 10" ; and the results are given to eight places : in Napier's canon
the intervals are 1', and the number of places is 7. The logarithms are strictly
Napierian, and the arrangement is identical with that in the canon of 1614.
This is the largest Napierian canon that has been calculated. The copy we
have seen is in the British Museum. In 1618 Ursinus published his
' Cursus Mathematicus,' of which there is a copy in the Cambridge Uni-
versity Library.

The only table of Napierian logarithms described in § 4 is Schitlze, 1778
[T. v.] (sines and tangents).

Art. 18. Logistic and Proportional Logarithms.

What are now called fractions or ratios used to be styled logistic numbers ;

and logistic logarithms are logarithms of ratios : thus a table of log -, ai

being the argument and a a constant, would be called a table of logistic or

proportional logarithms ; and since log - = log a — log x, it is clear that the

tabular results only difier from those of an ordinary table of logarithms by the
subtraction of a constant and a change of sign. , It appears that Kdplbe, in
his ' Chilias ' described below, originated tables of this Idnd ; but the step that
separates logistic from common logarithms is so smaU that no great interest

attaches to their first appearance. The use of the tabulation of log - in the

T^orking of proportions in which the thjrd term is a fixed quantity a is evident..

74 REPORT— isra.

There seems a tendency to keep the name logistic hijarithns for those tables
in which a = 3600" = 1° (so that the table gives log 3600 — log .^■, x being
expressed in minutes and seconds), and to use the naxae proportional logarithins
■when a has any other value. We have not met with any modern table of
this kind forming a separate work ; and such tables are usually of no great
extent. They are to be found, however, in many collections of tables ; and the
logistic logarithms from Callei were published separately at Nuremberg in
a tract of 9 pp. in 1843 (see title in § 5).

It may be remarked that tables of log - often extend to values of x

X

greater than a ; and then, in the portion of the table for which this is the
case, the mantissse are rendered positive (by the supposed addition of the
characteristic — 1, which is omitted) before tabulation.

Kepler, 1624. We cannot do better than foUow De Morgan's example,
and give a specimen of the table, which contains five columns : — •

53- 36-36

5-48

80500-00

19- 19-12

21691-30
124-15

48-18

The sinus or nuynerus absohitus is 805, which (to a radius 1000) is the
svas of 53° 36' 36", and the Napierian logarithm is 2169130. The third and
fifth columns are explained as follows : — if 1000 represent 24'*, then 805
represents ig"" 19"" 12= ; and if 1000 represents 60°, then 805 represents
48° 18' ; there are interscript diflferences for the first and fourth columns.
Thus, as De Morgan remarks, Kepler originated logistic logarithms. Kepler's
tract is reprinted by Maseres in vol. i. of his ' Scriptores Logarithmici '
(1791); and there is also, reprinted there " Joannis Kepleri .... supple-
mentum chiliadis logarithmorum . . . .Marpurgi, 1625," the original of which
we have not seen, but it contains no table. The copy of the 1624 work wo
have described is in the Cambridge University Library. For an account of
Kepler's ' Tabute Eudolphinaj,' see De Morgan.

Proportional logarithms for every second, a being 3°, are given almost
invariably in collections of nautical tables, usually to four places, but some-
times to five. T. 74 of Eaper, so frequently referred to in § 4, is a four-
place table of this kind, and was, as we have seen stated in several places, first
computed by Maskelyne. The reference was made to Eaper rather than
to any other of the numerous places where it occurs, as his work on
Navigation is one of the best-known, and has been through numerous
editions. Prof. Everett (Phil. Mag. Nov. 1866) says, quoting Eaper, that
proportional logarithms as at present used are a source of perpetual mis-
takes even to expert computers ; but this must be intended to apply
rather to practical men, as for the mathematical calculator they are very
convenient.

The following is a list of tables on the subject of this article, which are
described more fuUy in § 4.

Logistic logarithms for every second to 1°, viz. log 3600 — log x. — (To 4
places) (Jaediitee, 1742 and (Avignon) 1770, T. III. (to 4800") ; Donsour,
1747, T. XXXVI. (to 4800") ; Schtoze, 1778 [T. IV.] (to 3600") ; Vega,
1797, Vol. n. T. IV. (to 3600"); Goebon, 1849, T. XXI. (to 8600");
Cailet, 1853 [T. XI.] (to 5280") ; Hnrroir, 1858, T. VII. (to 5280") ;
Inman, 1871 [T. I.] (to 3600", intervals of 2").

Proportional logarithms for every second to 3°, viz. log 10,800 — log x. —
(To 5 places) Bios, 1809, T. XIV. ; Lai, 1821, T. XIV. ; Galbeaith,

ON MATHEMATICAL TABLES. 75

1827, T. X. ; BActiT, 1829, T. XXII. ; CotEMAisr, 1846, T. XXIV. ; IifMAif,
1871 [T. II.]

(To 4 places) (viz. T. 74 of Eapek) Cbosweli,, 1791, T. V. ; Maskeltne
(Eequisite Tables), 1802, T. XV. ; Bowditch, 1802, T. XV. ; Aotkew, 1805,
T. XIV. ; Mackay, 1810, T. LI. ; Mooke, 1814, T. XXV. ; Dtjooit, 1820,
T. VII. ; Keeigan, 1821, T. XII. ; Siambukt, 1822 [T. II.] ; Riddle,
1824, T. XXIX. ; J. Tayiob, 1833, T. XXXVI. ; Bbvert,bt (1833 ?), T.
XVIII. ; NoKTB, 1836, T. XXXIV.; Gkegoky &c., 1843, T. VIII. ; Guiefin,
1843, T. 41 ; J. Taylor, 1843, T. 35 ; RUMKEii, 1844, T. XXIV. ; Gordon,
1849, T. X. ; DoMKE, 1852, T. XL. ; Thomson, 1852, T. XIX. ; Rapbe,
1857, T. 74.

Proportional logarithms for every minute to 24'', viz. log 1440 — log x. —
(To 5 places) Galbeaith, 1827, T. IX.

(To 4 places) Stansbitky, 1822, T. G; Lynn, 1827, T. E; Gregory &c.
1843, T. XII. ; Gordon, 1849, T. XIX. ; Thomson, 1852, T. X. ; Eapek,
1857, T. 21A.

Art. 19. Tables of Gaussian Logaritlims.

Gaussian logarithms have for their object to facilitate the finding of the
logarithms of the suni and difference of two numbers whose logarithms are
known, the numbers being themselves unknown ; on this account they arc
often called Addition and Subtraction logaritlims. The problem is therefore •
given log a and log 5, find log (a + 6) by the taking out of only one logarithm.
The utility of such logarithms was first pointed out by Leonelli, in a very
scarce book printed at Bordeaux in the year XI. (1802 or 1803), under the
title " Supplement logarithmique ;" but it met with no success. Leonelli's idea
was to construct a table to 14 places — an extravagant extent, as Gauss has re-
marked. The first table constructed was calculated by Gauss, and published
by him in vol. xxvi. (p. 498 et seq.) of Zach's ' Monatliche Correspondenz '

(1812) : it give.s B and C for argument A, where A = log a;, B = log I ^ + - I

C = log (1 + .v), so that = A + B ; and the use is as follows. We have
identically —

log (a + 6) = log a + log (l + -\

^ log a + B I for argument log - j.

The rule therefore is, to take log a, the larger of the two logarithms,
and to enter the table with log a — log 6 as argument, we then have
log (a + 6) = log a + B, or, if we please, = log 6 + 0. Tor the difference,
the formula is log (a — i) = log 6 + A (argument sought in column C) if

log a log b is greater than "30103, and = log 5 — A (argument sought in

column B) if log a — log b is less than -SOlOS ; there are also other forms.
Gauss remarks that a complete seven-figure table of this kind would be very
useful. Such a table was formed by Matthiessen ; but the arrangement is
such that very little is gained by the use of it. This Gauss has pointed out
in No. 474 of the ' Astronomische Nachrichten,' 1848, and in a letter (1846)
to Schumacher, quoted by De Morgan. Gauss's papers on logarithms and
reviews of logarithmic tables from the ' Gottingische gelehrte Anzeigen,'
' Astronomische Nachrichten,' &c., are reprinted together on pp. 241-264 of
t. iii. of his ' Werke,' 1866. Of these pp. 244-252 have reference to Gaussian
logarithms and contain re^'iews of.PAswiQH, 1817 (§. 4), and -Matthiessen,

76 KEPOKT — 1873.

1818 (below). The largest tables are Zech (reprinted from Huisse's edition
of Vega) and WiTisiEijf, which answers the purpose Gauss had in view the
best of all : there is also a good introduction to the latter (in French and
German), explaining the iise and objects of the tables.

Whenever in this lleport the letters A, B, C are used in the description
of Gaussian logarithms, they are always supposed to have the meanings
assigned to them by Gauss (which are explained above), unless the con-
trary is expressly stated. Of course all Gaussian tables have reference to
Briggian (not hyperbolic) logarithms.

Iieonelli, 1806. This is the German translation of Leonelli's work, and
suggested to Gauss the constrnction of his table in Zach's ' Correspondenz.'
The book consists of two parts : in the first there are 9 pages of tables &o.
wanted in the construction of logarithms, viz. log x, log 1-x, log (1-O.r), . . . .

log (1-OOOOOOOOOO.r), for .■» = 1, 2, 9, to 20 places, and the same for

hyperbolic logarithms; also log 4, '2 (9-9), and log l-0,j', log 1-OOO.r,

log l-OOOOOa;, and log 1-OOOOOOOa;, for x = 01, 02, 99.

The second part is headed " Theorie der Erganzungs- und Verminderungs-
Logarithmen zur Berechnung der Logarithmen der Summen und Diflferenzen
von Zahlen aus ihren Logarithmen," and on pp. 52-54 the specimen table is

given ; log x being the argument, it gives log j 1 + - j and log (1 + x) as

tabular results to 14 places, for arguments from -00000 to "00104 at
intervals of -00001. [It wiU bo noticed that the above are the same as
Gauss's A, B, and C] The middle page of this table (p. 53) is nearly an
inch longer than any of the other pages of the book. The original work,
according to Houel, 1858, ' Avertissement,' p. vi, was published at Bordeaux,
An XI., under the title " Supplement logarithmique," ifeo.

Gauss, 1812. B| = log fl + i'jY and C (= log (1 + a;)) are given for

argument A(= log x) from A = -000 to 2-000 at intervals of -001, thence
to 3-40 at intervals of -01, and to 5-0 at intervals of -1, all to 5 places, with
diiferences. The table occupies 27 small octavo pages. Gauss's paper is re-
printed from the ' Correspondenz ' in t. iii. pp. 244-246 of his ' "Werke,'
1866 ; but the table is not reproduced there.

Matthiesseu, 1818. B and C are given to 7 places for argument A,
from A = -0000 to 2-0000 at intervals of -0001, thence to 3-OQO at intervals
of -001, to 4-00 at intervals of -01 and to 5-0 at intervals of -1 ; also for
A = 6 and 7, with proportional parts.

As C = A + B, the last three figures are the same for B and C, so that
the arrangement is, column of A, column of first four figures of B, column of
first four figures of C, column of last three figures of B and C, proportional
parts ; the eye has therefore to look in two different columns to take out a
logarithm. There is also another disadvantage, viz. that as there arc only
four figures of argument, if it is to be used as a seven-figure table three more
must be interpolated for.

The introduction is both in German and Latin.

Mr. Gray, who recalculated a considerable portion of this table, found that
it contained numerous errors (see Gkay, 1849, below). See also the intro-
ductory remarks to this article.

Weidenbacli, 1829. Modified Gaussian logarithms. Log x (= A) is

the argument, and log '^ "^ (= B) is the tabular result. A and B are thus

ON MATHEMATICAL TABLES. 77

"reciprocal, "the relation between them being, infact,10'^+^ = 10'^ + 10" + 1,
so tliat either A or B may be regarded as the argument. The table gives B to
hve places with differences, from A = -382 to A = 2'002 at intervals of -001,
from A = 2-00 to A = 3-60 at intervals of -01, and then to 5-5 at intervals
of 4. The commencement of the table being at A = -382 does not render it
incomplete, by reason of the reciprocity referred to above, since for arguments
less than -382 we can take B as the argument. Thus, at the beginning of
the table A and B are very nearly equal, viz. A = -382, B = 0-38355 ;
A = -383, B = •38255. There is an introduction of 2 pp. by Gauss.

The use of the table in the solution of triangles is very apparent, e.g. in

the formula cot - = -— . tan — !j^ — , in Napier's analogies, &e.

Gray, 1849. Modified Gaussian logarithms. T. I. Log (1 + .r) is the
tabular result for log x as argument ; and the range is from log x = -0000
to 2-0000 at intervals of -0001, to 6 places, with proportional parts to
hundredths (viz. 100 proportional parts of each difference).

T. IE. Xog (1 — .r) is the tabular result for log x as argument; and the
range is from log x = 3-000 to 1-000 at intervals of -001, and from 1-0000
to 1-9000 at intervals of -0001, to 6 places, with complete proportional parts.
The first table might have been copied from Mamhiessen by contracting the
7 places of the latter to 6 ; but it was recalculated by Mr. Gray, and many
errors were thereby found in Matthiessen's table (Introduction," p. vi) ; the
second table was also the result of an original calculation. Some remarks
and references on the subject of Gaussian logarithms &c. will be found in
the Introduction to the work.

Since writing the above account, Mr. Gray has sent us a copy of his
' Addendum to Tables and Formulse for the computation of Life Contin-
gencies .... Second Issue, comprising a large extension of the principal
table . . . . ' London, 1870, 8vo (26 pp. of tables and an introduction), which is
a continuation of the work under notice, and is intended to be bound up with it,
a new title having reference to the whole work when so augmented being addedi
The ' Addendum ' contains a table of log (1 + .r) to 6 places for argument
log a?, from log x = 3-000 to I-OOO at intervals of -001, and from 1-0000 to
00500 at intervals of -0001, the latter portion having proportional parts for
every hundredth of the dififerences added : the whole of course the result of
an original calculation. Mr. Peter Gray was the first to perceive the utility
of Gaussian logarithms in the calculation of life contingencies, and to him is
due their introduction as well as the calculation of the necessary tables, which
it is evident are valuable mathematically, apart from the particular subject
for which they were undertaken.

Zech, 1849, Table of seven-figure Gaussian logarithms. Denoting,

as was done by Gauss, log x, log (1 -t- -j, and log (1 + x), by A, B, C

respectively, then the table gives B to seven places, from A = -0000 to
A = 2-OOOOatintervalsof •0001,from A = 2-OOOto A = 4-000 at intervals of
•001, and thence to G-00 at intervals of -01, with proportional parts through-
out ; the whole arranged as an ordinary seven-figure logarithm table, and
headed Addition table.

The Subtraction table gives C to 7 places, from B = -0000000 to -0003000
at intervals of -0000001, thence to -050000 at intervals of -000001, and
thence to -30300 at intervals of -00001 to seven places, with proportional
parts.

78 REPORT— 1873.

, The addition table occupies 45 pp., the subtraction table 156 pp. The
whole is a reprint from Hulsse's Vesa of 1849, the paging being unaltered,
and running from 636 to 836. The second edition is identical with the first,
except that the 3 pp. of introduction are omitted.

Wittstein, 1866. A fine table of Gaussian logarithms in a modified
form. B (=log (1 +a;)) is given to seven places for the argument A ( =Iog a')
for values of the argument from 3-0 to 4-0 at intervals of •!, from 4*00 to
6 00 at intervals of -01, from 6-000 to 8-000 at intervals of -001, from
80000 to lOOOOO at intervals of -0001, and also from -0000 to 4-0000 at the
8a:me intervals. Differences and proportional parts (or rather multiples) are
given, except ou one page (p. 5), where they are given for. alternate
differences as there is not sufiicient space.

The arrangement is similar to that of a seven-figure logarithmic table.
The figures have heads and tails, and are very clear.

Ou p. 127 there is given a recapitulation to three places, and to hundredths,
of part of the table and the formula). A complete explanation is given in
the introduction to the work.

Gaussian logarithms are very useful in the solution of triangles'in such

formulae as cot q = _ . tan (A— B), in which "Weidenbach's table would

also be useful.

The following is a list of tables of Gaussian logarithms contained in
works noticed in § 4.

Tables of Gaussian logarithms. — PAsaTJicn, 1817, T. III. (5 places) ;
[Encice, 1828] [T. III.] (4 places) ; X6hi,ee, 1832 [T. III.] ; HtassE's Vega,
1840, T. XII. ; MtJLLEE, 1844 [T. II,] ; [Sheepshanks, 1844] [T. V.] ;
KoHLEu, 1848 [T. II.] ; Shokthede, 1849, T. VII. ; Filipowski, 1849, T, II. ;
HotEi,, 1858, T. III. ; Galbbaiih and Haughion, 1860 [T. IV.] ; Oppolzeb,
1866.

Art. 20. Tableg to convert Briggian into By^erholic Logarithms, and vice versd.

Tables for the conversion of Briggian into 'hyperbolic logarithms, and vic^
vend, are given in nearly all collections of logarithmic tables. Such a table
merely consists of the first hundred (sometimes only the first ten) multiples

of the modulus -43429 44819 03251 82765 11289 , and its reciprocal

2-30258 50929 94045 68401 79914 , to five, six, eight, and ten or even

more places. A list of such tables, contained in works described in § 4, is
given below ; tables of this kind, however, rarely exceed a page in extent,
and are very easy to construct. It is not unlikely that the list is far from
perfect, for in some cases it was not thought worth while noticing such
tables when of small extent and to few places. We mention Degen (§ 4) as
containing one of. the largest.

The following is a list of tables contained in works noticed in § 4.

To convert Briggian into hyperbolic logarithms and vice versd. — (To more
than 10 places) Schuizb, 1778 [T. I.] ; 'Degen, 1824, T. II. ; Shouieede,
1849, T. VII. ; Callet, 1853 [T. IV.] ; Paekhukst, 1871, T. V.

(To 10 places) Schboit, 1860, T. I, ; Beuiins, 1870.

(To 8 places) Shoetbedb (Tables), 1844, T. XXXIX. ; Kohleu, 1848,
[T. I.] ; HoUEL, 1868, T. III.

(To 7 places) Beemikee, 1862, T. I. ; Beemikee's Vega, 1857, T. I. ;
DupuM, 1868, T. V.

(To 6 places) Dobson, 1747, T. XXXVII.

ON MATHEMATICAL TABLES. 79

(To 5 places) De Pkasse, 1814 [T. II.] (?) ; Gallr.\ith and HATJGnioir;
1860 [T. I.] ; Wackerbamh, 1867, T. V.

See also Trotter, 1841 [T. I.]; SenLOMiLCH [1865 ?] ; Eankike, 1866^
T. 3 ; and Piijeto, 1871 (§ 3, art. 13).

Art. 21. Interpohtton Tables.

All the tables of proportional parts (described in § 3, art. 2) are
interx)olation tables in one, and tbat the most nsnal, sense ; and similarly^
multiplication and product tables may be so regarded (see § 3, art. 2). We
may, however, especially refer to Schron, 1860, as its printed title describes
it as an interpolation fable — a designation not common. The only separate
table we have seen for facilitating interpolations, -when the second, third, &c.
differences are included, is Woolho¥se, noticed below. We may also refer
to Godward's tables (title in § 5), but they seem of such special application
that we have not thought it necessary to describe their contents.

■Woolliouse, 1865. Papers extracted from vols. xi. and xii. of the
' Assurance Magazine.' There are 9 pp. of interpolation tables (viz. pp.
14r-22) . The work contains a clear explanation of methods of interpolation,
with developments.

The following are references to tables described in § 4.

Binomial-theorem eoijkients. — Schulzb, 1778 [T. XIII.] ; Vega, 1797,
Vol. II. [T. VI.] ; Barlow, 1814, T. VII. ; Hantschl, 1827, T. IX. ;
Hulsse's VEtti, 1840, T. XIII.-; Kohler, 1848, T. X.; Parkhursi, 1871,
T. XXXII. See also Eottse (§ 3, art. 25).

Other interpolation coefficients. — Peters, 1871 [T. IV.], I. and II.

Coefficients of series terms. — Vega, 1797, Vol. II. [T. VI.] ; Lambert, 1798,
T. XLIV. ; HtrissE's Vega, 1840, T, VIII. ; Kohler, 1848, T. XI.

Art. 22, Mensuration Tables.

We have made ho special search for tables on mensuration (such as areas
of circles of given radius, volumes of cones of given base and altitude, &c.),
and have only included those that have fallen in our wa/ in the course of
seeking for more strictly mathematical tables during the preparation of this
Eeport. As, however, for several reasons it seems desirable that a complete
list of such tables should be formed, we shall endeavour to render this
Article as nearly perfect as we can in the supplement. One reason, how-
ever, why such tables are not of very high mathematical value is that the'
measures arc generally expressed in more or less arbitrary units, such as yards,
feet, inches, &c., or metres &c.

We may especially refer to the large table of circular segments in Sharp,
1717 (§ 4).

Sir Jonas Moore (1G60?). The table is a very small one, and
scarcely occupies a third of a folio page. It gives the periphery of an
ellipse for one axis as argument (the other axis being supposed eqital to
Unity) to 4 places, with differences ; the range of the argument is from -00
to 1-00 at intervals of -01. Thus, to find the perimeter of an ellipse, axes 1
and -78, we enter the table at 78 and iind 2-8038. If one axis is not equal
to unity, a simple proportion of course gives the perimeter. After working
out four examples, the author proceeds : " I have made above 45,000 arith-
metical operations for this table, and am now well pleased.it is finished.'

80 UEPOM— 1873.

Some perhaps may fiad shorter ways, as I believed I had myself, till advised
otherwise by the truly Honourable the Lord Bruncher, ifec." This is perhaps
the first tabulation of an elliptic integral.

Bomnycastle, 1831. A table of the areas of segments (pp. 29.5-300) :
the same as T. XIII. of HajtISCtil.

Todd, 1853. T. I., Areas (to 6 places) and circumferences (to 5 places)
of circles for the diameter as argument, the range being from diameter jL,-
to diameter 24 at intervals of yL; the decimal fractions (to 4 places)
equivalent to y'g-, -^^, &c., are printed at the top of each page.

T. II. The same from diameter 24 to 100 at intervals of A (4 places
only for the circumferences).

.T. III. The same from diameter 12 to 600 at intervals of unity. Both
areas and circumferences are only given to 4 places.

T. IV. The same from diameter -1 to' 100 at intervals of -1. Areas to 6
places, circumferences to 5.

T. V. to VII. stand in exactly the same relation to spheres that T. I. to
IV. do to circles, except that T. V. is equivalent to T. I. and II., the
intervals being J from 1 to 100 ; and T. VI. commences at 1 (not 12). The
volumes and superficies are given to 4 places.

T. VIII. Areas (exact) and diagonals (to 5 places) of squares for side as
argument, from A to 100 at intervals of ^.

In all cases the arguments are given in inches, and the results in square
and cubic inches ; but in T. III. and VI. the corresponding numbers of
lineal", square, and cubic feet are also given.

The original work, of which this is the second and greatly augmented
edition, was published in 1826 ; and the tables were the result of original
calculations. There are besides some specific gravities, &c.

The following tables are more fully described in § 4.

Mensuration tables. — Sharp, 1717 [T'. II.], areas of segments of circles ;
[T. III.], table for computing the solidity of the upright hyperbolic section
of a cone ; Dodson, 1747, T. XXVI., XXVIII., and XXIX. ; Gamkaitit,
1827, T. XV. and XVI. (Introd.) ; Haktschl, 1827, T. XIII. ; Teotiek,
1841 [T. v.] and [T. XII.]; WiiLtcu, 1853, T. (circumferences and areas
of circles); Bba4dmoee, 1862, T. 34 (circumferences and areas of circles);
Eakkike, 1866, T. 4 and 5.

Art. 23. Dual Logarithms.

Dual logarithms were invented, and the tables of them calculated, by ISIr.
Oliver Byrne, who, besides the work described below, has published ' Dual
Arithmetic ' and the ' Young Dual Arithmetician ' on the subject. A dual
number of the ascending scale is a continued product of powers of 1*1, l-O],
1-001, &c!, taken in order, the powers only being expressed. To distinguish
these numbers from ordinary numbers, they are preceded by the sign \\/:
thus, \|/ 6, 9, 7, 6 = (1-1)» (1-01/ (1-001)' (1-0001)» ; n,|/ 0, 0, 2 = (1-1)°
(l-Ol)" (1-001)^, the numbers following the \|/ being called dual digits.
"When all but the last digit of a dual number are zeros, the dual number is
called a dual logarithm ; but the dual logarithms used by Mr. Byrne are " of
the eighth position," viz. there are 7 ciphers between the \|/ and the
logarithm.

A dual number of the descending branch is a continued product of powers
of -9, -99, -999, &c., and the dual number is followed by the symbol /|\ :
thus, (-9)3 (-99)2 = .3 '2 /|\; (-999)' (-999999)2 = '0' 0' 3' 0' 0' 2 /|\. In the
descending branch also a dual number reduced to the eighth position is.

ON MATHEMATICAL TABLES. 81

called a dual logarithm, and is to be considered negative if the ascending
dual logarithm is taken positive, and vice vend.

ByrnCj 1867. T. I. contains all the dual numbers of the ascending
branch of dual arithmetic from \|/ 0, 0, 0, 1 to \|/ 7, 3, 1, 9, and their
corresponding dual numbers and natural numbers. The range of the dual
logarithms is from 00000 to 69892175, and of the natural numbers from
1-00000000 to 2-01167234. Marginal tables are added, by means of which
all dual numbers of 8 digits, and their corresponding dual logarithms and
natural numb'ers, may be derived : the table occupies 74 pp.

T. II. Dual logarithms and dual numbers of the descending branch of
dual arithmetic from '0 '0 '0 '1 '0 '0 '0 '0 /|\ to '3 '6 '9 '9 '0 '0 '0 '0 /|\ with
corresponding natural numbers. The range of the dual logarithms is from
'10001 to '39633845, and of the natural numbers from -99990000 to
•67277805. Marginal tables are added, by means of which aU intermediate
dual numbers of 8 digits and their corresponding dual logarithms and natural
numbers may be derived. This table is printed in red, T. I, and III. being
in black. It occupies 38 pp.

T. III. Natural sines and arcs to 7 places for every minute of the
quadrant. The length of the arc is, of course, the circular measure of the
angle, so that we have a table of circular measures to minutes : the arrange-
ment is quadrantal. Proportional parts are given for 10", 20". . . .90" for
each difference ; and these occupy two thirds of the page. There are small
proportional-part tables for the arc : the table occupies 90 pp.

The author claims that his tables are incomparably superior to those of
common logarithms, and asserts that " these tables are equal in power to
Babbage's and CaUet's, and take up less than one eighth of the space "
('Dual Arithmetic,' part ii. p. ix). Bahhage and Collet seems an error
(unless the Callbt of 1827 (§ 3, art. 15) is meant), as the latter work con-
tains the table of the logarithms of numbers which is the sole contents of the
former. Mr. Byrne's works on the subject are : — ' Dual Arithmetic : a new
Art,' London, 1863, 8vo (pp. 244) ; ' Dual Arithmetic : a new Art. New
Issue, with a complete analysis,' 1864 (pp. 83) [this work contains a table
of 3 pp., " to facilitate the conversion of dual numbers into common ones, or
the converse "] ; ' Dual Arithmetic: a new Art. Part the Second ' (pp. 218),
and the work above described. Mr. Byrne has also published ' The Dual
Doctrine of Angular Magnitude and Functions, &c.,' and the ' Young Dual
Arithmetician,' neither of which we have seen: the latter contains an
abridgment to 3 dual digits of the tables in the work described above.

In spite of the somewhat extravagant claims advanced by the author for
his system, dual logarithms have found but little favour as yet either from
mathematicians or computers.

Art. 24. Mathematical Constants.
In nearly all tables of logarithms there is a page devoted to certain
frequently used constants and their logarithms, such as tt, -, tt^, ^/tt, a /g,

&c. the radius of the circle in degrees, minutes, &c., the modulus &c.
There are not generally more than four or five logarithms involving n given ;
and usually half the page is devoted to constants' relating to the conversion
of weights and measures. It is only necessary, therefore, here to refer to
works in which there is a better collection than usual of constants.
1873. •*

82 REPORT— 1873.

A very good colleotion is given by Matnaed (described' below), and
also by Btene, 1849. This portion of the present Keport is very far from
complete, as the values of mathematical constants have, as a rule, appeared in
periodical publications, while those only that are most used by the general
computer are to be found in collections of mathematical tables. We refrain,
therefore, from giving references to several periodicals we have met with
containing constants, as they belong properly to a subsequent portion of the
Report ; and it is hoped that, after the completion of the examination of
the memoirs, a pretty complete list, either of the constants themselves, or at
all events of the places where they are to be found, will be given.

We may, however, notice a paper of Paucbei-'s in the first volume of
' Grunert's Archiv der Mathematik und Physik,' in which a number of
constants involving ir are given to a great many places, and Gauss's
memoirs on the lemniscate-functions (' Werke,' t. iii. pp. 426 &c.)j where

e~^, e~*^, e~^^, &c. are calculated to about fifty places. On Euler's con-
stant, see ' Proceedings of the Boyal Society,' t. xv. p. 429 ; t. xvi. pp. 154,
299 ; t. xviii. p. 49 (Shanks) ; t. xix. p. 514 (Glaisher) ; t. xx. pp. 27, 29
(Shanks). On e, the base of the Napierian logarithms, log,2, log,3 &o., see,
besides the places just referred to, ' Eoy. Soc. Proc' t. vi. p. 397, and ' Brit.
Assoc. Eeport ' (Sections) 1871, p. 16, and also Shanks 1853 (§ 4). Several
constants are to be found in the different works of Maseres. Mr. Maynard
and Mr. Merrifield have independently calculated log.M and log,OT (M and m
being the modulus and its reciprocal) to 30 places (' Assurance Magazine,'
t. vi. p. 298).

The value of tt has been calculated to 500 places of decimals by Shanks
and Eichter independently, and to 707 places by the former alone : see
references, ' Messenger of Mathematics,' December 1872 and July 1873. Mr.
Shanks's latest value appears in the 'Eoy. Soc. Proc' t. xxi. p. 319.
It is proper here to remark that Eutherford's 208-decimal value of ir, given
in the ' Phil. Trans.' 1841, p. 283, is erroneous after the 152nd place : this
value is reproduced in BrKWE, 1849 (§ 4), and in Matnakd ; so that it is
erroneous also in both of these works.

[Maynard.] A good table of constants involving tt, such as tt ^2, tt "',
Vt, &o., and some few involving e &c., to a great many (generally 30)
places. There are also other constants not included in the subjects of this
Eeport.

The^Qppy of these constants that we examined consisted of six leaves
without axover, and which were evidently extracted from some work. Mr.
C. W. Merrifield, E.E.S., subsequently called our attention to the particu-
larly good dollection of constants in ' The Millwright and Engineers' Pocket
Companion;. ... By Witliam Templeton. . . . Corrected by Samuel May-
nard .... Pifteenth edition, carefully revised,' London, 1871,^8vo, and lent
us a copy ; and on examination it appeared that it was to this work that
Maynard's collection belonged, where it occupies pp. 169*180. There are,
altogether, 58 constants involving t, and their logarithms, given generally to
30 places, and 13 others that may also be properly styled mathematical. It
is mentioned that part of the table had previously appeared in Keith's
' Measurer ' (twenty-fourth edition, 1846). Templeton's work contains several
other tables (areas of circles, &c.), and square roots which would have been
-included in this Eeport had we seen the book earlier ; as it is they will be
noticed in the Supplement. On Eutherford's value of tt, quoted by May-
nard, see introductory remarks to this article.

ON MATHEMATICAL TABLES. 83

The following is a list of references to § 4.

Lists of Constants.— DcmsQTS, 1747, T. XXVII. ; Gaibeaith, 1827, T.
LXIII. ; Hantschl, 1827, T. XI. ; [Da Moegan], 1839 [T. V.] ; Faeiet,
1840 [T. III.] ; MtaLEE, 1844 [T. IV.] ; Shokteede (Tables), 1844, T. II.
MttoEE, 1844 [T. IV.] ! Eapbe, 1846, T. V.; Xohlbe,' 1848 [T. III.]
BrKNE, 1849 [T. III.] ; Beemikee, 1852, T. II. ; Willich, 1853, T. XX., &c.
Shanks, 1853 (constants to a great many places) ; Beemikee's Vesa, 1857
HoOei,, 1858, T. Vni. ; Htttion, 1858, T. XII. ; Galbkaith and HATreHTON,
1860 [T. IV.]; "Wackeebakth, 1867, T. IV., V., and XXI. ; BETrmra, 1870.

Note. — Binomial-tlieoreni coefficients and coefficients of series-terms are
noticed under Interpolation Tables in § 3, art. 21.

Art. 25. Miscellaneous Tables, flgwrate Numbers, Sfc,

"We have placed in this article tables which could not properly be
described under any one of the previous twenty-four heads. The list is jiot,
however, a long one, as we have frequently placed doubtful tables in the
article which most nearly applied to them.

We may refer especially to JoircotrEi's table of triangular numbers (de-
scribed below), which is perhaps unique. Betshammee's commercial loga-
rithms and Montpeeeiee's binary logarithms are described in § 3, art. 13.
Picaete's table to facilitate the performance of divisions is described in § 3,
art. 7. We may also particularly notice De&bn's large table (§ 4) of log
1-2. . . .a;). There is a table of binomial-theorem coefficients in EotrsB (see
below) ; and other tables of the same kind are referred to under Interpolation
Tables in § 3, art. 21. Tables of endings of squares are noticed in § 3,
art. 4 ; and tables for the solution of cubic equations, viz. + (a; — x'), in
§ 3, art. 5.

Browne, 1731. Pp. 6 and 7 are occupied by a table headed " Area of

the circle in. degrees and to the 10,000th part of a degree," Calling ^, a,

it gives o, 2a, 3a 100a, 200a, 300a, and 360a to 7 figures. There are

also three other columns in which the results only differ by a change of
decimal point.

Through a mistake in the printing in the copy before us, all the odd pages
are upside down.

Heilbronuer, 1742. On pp. 922-924, the numbers from unity to 140,
72, and 100 are expressed in the scales whose radices are 3, 2, and 12
respectively.

Joncourt, 1762 [T. I.]. A table of triangular numbers up to that of

20,000, viz. .^ ^" "^ ^^ for all numbers from w = 1 to 20,000 (the table

occupies 224 pp.).

fT. II.] Cubes of numbers from 1 to 600.

There are 36 pages of explanation &o., in which it is shown how [T. I.]
may be used in the extraction of square roots, &c, De Morgan refers to this
book as " De la Nature . . . . de Nombres trigonaux," 1762, so we suppose
some copies vnth the introduction &c. in French were published. The
Eoyal Society's copy has , " Deo. 23, 1762," written in ink underneath the
printed date. The book ia handsomely printed.

The Babbage Catalogue also gives the same work with an English title.
' The Nature and Notable Use of the most simple trigonal numbers, with

g2

84 REPoiiT — 1873.

two additional tables, &o., translated from the Latin of E. de Joncourt by
the author's self.'

Phillips, 1829. This is not properly a table at all. Names and an
abbreviated way of writing them are suggested for all numbers up to 9
followed by 4000 figures, the chief peculiarity of the system being that 1000
is called ten hundred, and 10,000 a thousand, and so on. The only
explanation of the object of the table is contained in the curiously untrue
remark that, by adopting the author's names, " we obtain a clearer view of
calculations which are generally called inconceivable only because we have
hitherto adopted no terms to express and limit them." On Sir R. Phillips,
and the value of his works, see De Morgan's ' Budget of Paradoxes ' (1872),
pp. 143-145.

D. Galbraith, 1838. A piece contains 4, 5.... 56 squares, and the
table is to show the number of dozens in any number of pieces up to 100,

&c. It contains f| for a; = 4, 5. . . .56, and y = l, 2, 3 100, 200,

300, 400, and 500, the value of x being constant over any one page : thus
a; = 15, 1/ = 65, we have given 81-3 for -jlg (15 x 65) = 81 Jj. The table was
calculated to give the number of handkerchiefs in any number of pieces, &c.

De IlSorgan, 1843. Dboen's table (§ 4) of log (1, 2 ... .a;) is reprinted
to six places by De Morgan at the end of his article on " ProbabiHties " in
the ' Encyclopsedia Metropohtana.' The last figure is not corrected : the
table occupies pp. 486-490,

Rouse (no date). The tables, which are neither elaborate nor very nume-
rous, are not of sufficient mathematical value to render it necessary to do more
than give a general idea of their contents. In the body of the work are a num-
ber of small tables of this kind : — A and B (of equal skill) play 21 games ; and
the odds in favour of A's winning 1, 2 ... . 20, 21 are given as tabular results.
Similar tables are given for 20, 19 . . . . 2 games played. Then we have the
same when the odds in favour of A are 6 to 5, 5 to 4, 5 to 3, &c., — the
maximum number of games, however, being six. On a folding sheet at the
end is given the number of ways in which 1, 2, 3 .... 60 points can be
thrown with 1, 2. . . .10 dice, and also the number of ways in which 52
cards can be combined into 4 hands in any given manner (thus, 5 diamonds,
4 hearts, 3 spades, and 1 club can be obtained in 3421322190 ways); the
factor and the result when the suits are not specified are also given. The
mode of formation of the table is obvious.

On a folding sheet at the beginning of the book is given (a + 6)" at
fuU length for m = 1, 2 .... 30.

The following is a list of miscellaneous tables contained in works that are
described in § 4. For greater convenience a brief description of the contents
of each table is appended to the reference to it.

MguraU Numbers. — Lambebt, 1798, T. XXXVII.

Hyperholie Antilogariihms (viz. powers of e) and their Briggian logarithms
— ScHTTLZB, 1778 [T. I.] ; VESA, 1797, Vol. II. T. ni. ; Lambbet, 1798, T,
XI. ; HtLssE's Vega, T. VII. ; Kohlee, 1848, T. III. ; Shoktrede, 1844
[T. n.], III. ; HtrrroN, 1858, T. XII. ; Cailet, 1853 [T. II.], HI.

Miscellaneous. — Sharp, 1717 [T. I.] | multiples of j\ ; Dodson, 1747,

T. XX. (combinations), T. XXIII. (permutations), T. XXXV. (seconds in any
number of minutes less than 2°) ; Sohulzb, 1778 (Pythagorean triangles) ;
Maseses, 1795 (multiples of primes) ; Vega, 1797, Vol. II. [T. VII.] and
[T. Vni.] (piling of shot) ; Lambbet, 1798, T. II. (multiples of primes), T.

ON MATHEMATICAL TABLES. 85

III. (products of consecutive primes), T. XVII. (numbers of the form
2.3»5,7,)^ T. XXiy. (f, 0\...for = 10,000" m, &c.), T. XXXII.
(Functiones hyperbolicsB circularibus analogee); Bokda and Delambre,
1800 or 1801 [T. V.] (log sin (x + 2)— log sin x, &c. centesimal) ; Peakson,
1824 [T. II.] (1°, 2°. . . .as decimals of the circumference) ; Deoen, 182.4,
T. I. (large table of log (1.2. . . .«)), T. Ill.-(multiple8 of log 2, log 3, &c.) ;
Ubsintts, 1827 [T. IV.] (length of chords subtending given angles) ; Hantschl,
1827, T. XI. (multiples of constants); HABTie,'1829 (contents of solids ex-
pressed in Fuss and ZoU) ; [DeMoesan], 1839 [T. VI.], (log (1.2.3 x))i

HtlissE's Ve&a, 1840, T. IV. (chord table), T. IX. i^and (x ^^, &c.) ;

Shokteede (tables), 1849, T. IV. and V. (for calculating logarithms and anti-
logarithms), and T. VIII. (log (1.2.3..,.^)); Domkb, 1852, T. XXX.

H '"» + ^ j j ; Shanks, 1853 [T. I.] (terms of tan ->i and tan -'^) ;

SoHEON, 1860, T, III. /hyp. log 10" and 1 + r^\; *ScHLomi,cH [1865?]

(elliptic quadrants); Evekeit, 1866; "Wackbebaeth, 1867, T. II. (log
(1.2.... a;), log (1.3.... a;), log (2.4 . . . . a;)) ; Parkhuest, 1871, T. IV.,
VI.-VIII., X., XI., XV.-XVn., XIX., XXIV., XXIX., XXXVI. Se-
also Ktjiik:, 1848, T. 2-10 and 11 (Theory-of-number tables and multiples of

TT and -\ (§ 3, art. 4).

§ 4. WorJes containing Collections of Tables, arranged in alphabetical order,
[The titles of the works can be found by reference to § 5.]

Academic de Prusse (1776). This collection of tables only containB
two that come within the scope of this Report.

[T. I.] (vol. iii. pp. 172-207). Table of sines, expressed as arcs whose
length is equal to that of the sine ; viz. for a; (expressed in degrees and mi-
nutes) as -argument there is given the angle (expressed in degrees, minutes,
seconds, and tenths of a second) whose circular measure is sin x, the argu-
ment X being given to every minute of the quadrant. There are no differ-
ences ; and the arrangement of the table is quadrantal (not semiquadrantal).
The table is due to Schulze.

[T. II.] (Vol. iii. pp. 258-271). Lengths of circular arcs, viz. the- circular
measures of 1°, 2°, 3°, . . . . 360°, of 1', 2',.. .. 60', and of 1", 2", .... 60" to
27 places. This table is by Schulze, in whose collection it also appears : see
SCHU-LZB [T. VII.].

Both these tables are included under the head " Tables auxUiaires " in the
third volume.

The whole work is attributed in the Eoyal Society's Catalogue to Schulze,
and, from internal evidence we have little doubt, correctly.

Adams, 1796 [T. I.]. Six-figure logarithms to 10,860, written at length,
with characteristics. Differences are added.

[T.. II.] Log sines, tangents, and secants for every minute of the qua-
drant, to 6 places ; with tables at the bottom of the page to facilitate inter-
polations.

[T. III.] Log sines, cosines, tangents, cotangents, secants, and cosecants for
every quarter point, to 5 places.

86 REPORT— 1873.

Andrew, 1806. T. XIII. Squares of natural semichords, viz. sin' g

from a; = 0° to a;=120° at intervals of 10", to seven places, -with differences
and proportional parts for seconds. This valuable table occupies pp. 29-148
of the work.

T. XIV. Proportional logarithms to 3°, at intervals of a second, to four
places ; same as T. 74 of Eapeb.
The other tables are nautical.

Anonymous [1860 ?]. Four-figure logarithms of numbers from 100 to
1000, with proportional parts, on a card (about 12 in. by 10 in.). _ On the
back, numbers (to four figures) to logarithms from -000 to 1-000, at intervals
of -001, with proportional parts. Printed by J. Sittenfeld, published by
Yeit and Co., Berlin. No date. The Brit.-Mus. copy received April 2, 1860.
Bagay, 1829. T. XXII. Proportional or logistic logarithms for every
second to 3° (or 3'') to five places ; same as T. 74 of Kapek, except to five
instead of four places.

T. XXIII. Seven-figure logarithms, from unity to 21,600 (with the cor-
responding degrees, minutes, and seconds), to seven places, with differences,
but not proportional parts.

T. XXrV. Logarithms of sexagesimal numbers, viz. logarithms of num-
bers of seconds in all angles from 6° 10' 0" to 12°, at intervals of 1", to five
places.

Appesbix. — Table of log sines and tangents for every second of the qua-
drant to seven places (without differences). The change in the middle of the
column is beautifully clearly marked by a large black nucleus, surrounded by
a circle, printed instead of zero. ' Only the first logarithm affected is so de-
noted ; but the mark is so striking that it readily attracts the eye. The table
was formed by interpolation from Cailet, corrected by Taylor (see p. ii of
the ' Avertissement ') ; 76 errors were thus found in Taylor. Some errata
are given at the end of the work.

All the other tables are astronomical. This work, Which has now become
rare, is much esteemed.

BarloW} 1814. T. I. Squares, cubes, square and cube roots (to 7 places),
reciprocals (to 9 places as far as 1000, afterwards to 10), and all factors of
numbers from 1 to 10,000, Thus, for the factors of 4932 we have given 2".
3". 137.

T. II. The first ten powers of numbers from 1 to 100. This table was
taken from Huiroif [T. TV.] and Vega (Tabulae), vol. ii. T. IV. The errors
given in this Keport in Hutton are not reproduced in this table.
T. III. Fourth and fifth powers of numbers from 100 to 1000.
T. rV. For the solution of the irreducible case in cubic equations ;. viz.
y^-y is tabulated from y=l-0000 to 1'1549, at intervals of -0001, to 8
places.

T. V. Prime numbers from 1 to 100,103 (this table is incorrectly described
on the titlepage to it as extending to i0,000 only).

T. VI. Hyperbolic logarithms, to 8 places, of numbers from unity to 10,000
(this table is incorrectly described on the titlepage to it as only extending
from 1000 to 10,000)

T. VII. Differential coeffieients, viz. the first six binomial- theorem coefii-
. ^ m(n— 1) n(n—l)....(n—S) „ „, , , „^

cients, ^^ ' ^' -^ — Y2 6 '' ™ w="01 to 1-00, at intervals of

•01, to 7 places.

ON MATHEMATICAL TABLES. 87

These tables occupy 256 pp., and are followed by 78 pp. of formulse, weights,
and measures, &c.

There is a full introduction, stating whence the tables were derived, or, if
computed, from what formulas, &o. The hyperbolic logarithms were taken
from "Wolfram's table in Schtozb ; and the reciprocals, factors, square and
cube roots, and several other ^tables were the result of independent cal-
culations.

The squares, cubes, square and cube roots, and reciprocals from this table
were reprinted and stereotyped, at the suggestion of De Morgan, in 1840 (see
Barlow's tables,. 1840, in § 3, art. 4). The reprint thus gives T. I., the
column of factors being omitted. A list of 90 errors in T. I. of the original
work is given in the reprint ; and 25 errors in T. VI. are given by Prof.
Wackerbarth in the ' Monthly Notices of the Eoyal Astronomical Society ' for
April 1867.

Bates, 1781. [T. I.] Five-figure logarithms to 10,000, without dif-
ferences.

[T. II.] Log sines and tangents (to 5 places), and natural sines and tan-
gents (to 7 places), for every minute of the quadrant, semiquadrantally
arranged ; no differences.

The tables (which have a separate titlepage, bearing the date 1779) are
preceded by 211 pp. of trigonometry, and followed by an Appendix on the
motion of projectiles in a non^resisting medium. The work was intended for
use in the Military Academy, Belmont, near Dublin.

Beardmore, 1862. Only 23 pages (pp. 84-106) of this work contain
tables that come within the scope of this E,eport.

T. 34. Areas and circumferences of circles, to 3 places, for diameters
•1, -2, -9, and from 1-00 to 100, at intervals of •25.

T. 35. Squares, cubes, fifth powers, square and cube roots (to 3 places),
"and reciprocals (to 9 places) for numbers from 1 to 100, the squares and
square and cube roots being given as far as 1100.

T. 36. Six-figure logarithms of numbers from 100 to 1000.

T. 37. Log sines from 0° to 45° 50', at intervals of 10', to 6 places.

T. 38. Natural sines, tangents, and secants for 1°, 2°, . . . . 90"^, to 6 places.
The other tables relate to hydraulics, rainfall, &c.

The work was first published in 1850 ; and a second edition, in an extended
form, was issued in 1851.

Beverley [1833?] T. VI. (p. 127). Any number of minutes less than'
12'' expressed as a decimal of 12'', to 4 places.

T. VI. (pp. 232-243), Sexagesimal cosecants and cotangents for every
minute from 20° to 90°. A sexagesimal cotangent is the cotangent when
the radius is taken = 60' (or 1°) ; viz. it bears to 60' the same ratio that the
ordinary cotangent does to unity, and is usually expressed in minutes, seconds,
and decimals of a second. The same, of course, holds for sines, cosines, &o.
Thus the sexagesimal sine of 30° is 80', cosecant 30° =120', &e.

In this table the quantities tabulated are not sexagesimal functions, but
sexagesimal functions divided by 3 (and are therefore to radius 20') : we thus
have cosec 30° =40'. The table is given to two decimal places of a second.

T. XV. Sexagesimal sines, tangents, secants, and versed sines (viz. to rad.
60') to every degree to 90°, to one decimal place of a second, with differences.

"T. XVII. Log sines and tangents, from 18° to 90°, at intervals of 1', to
4 places.

T. XVIII. Proportional logarithms for every second to 3°, to 4 places ;
same as T. 74 of Eapj;k.

88 REPORT — 1873.

Mr. Beverley made some improvements in Tatloe'S Sexagesimal Table
(§ 3, art. 9), and devised a plan to introduce them into Taylor's table without
reprinting it. He accordingly made application to the Board of Admiralty to
be allowed to do so in the copies that remained unsold ; but this was refused.
He then offered to purchase all the unsold copies of Httttow's ' Products '
and Tatloe's tables, in order to introduce his improvements ; but his applica-
tion was refused after the terms had been agreed upon, because he asked for
six months' credit. In the Appendix he complains that " the immense
labour that the calculation of his tables required him to exert had then ruined
his constitution, and brought him to the verge of a premature grave." It is
to be presiimed that the Admiralty had some grounds for their refusal ; but
it is certain that no use has been made of Hutton or Taylor since the time of
Mr. Beverley's application. No pains at any time seem to have been taken
to circulate or make known any of the books published by the Board of
Longitude, so that none of them have ever come into general use.

Mr. Beverley died in 1834, at the age of 39 ; and the present work was
published after his death, as it contains a notice of his life by " J. B.", and
evident traces of revision. He often refers to his Taylor's Sexagesimal Table,
but no doubt it was never published. We have seen ' The Book of Formulae
&c., Cirencester, 1838,' by the same author ; but it contains no tables.

Borda and Delambre, An IX. (1800 or 1801). [T. I.] Seven-figure
logarithms of numbers from 10,000, to 100,000, with differences and pro-
portional parts for all. The line is broken when a change takes place in
the middle of it. It may be remarked that while in aU modern tables
of logarithms of numbers three figures are common to the block, and
four only are given in the columns, in this table there are but two leading
figures, and five are found in the columns, so that the lines are broken in
very few instances. [T. II.] Eleven-figure logarithms of numbers to 1000,
and from 100,000 to 102,000 (the latter with differences).

[T. III.] Log sines, cosines, tangents, and cotangents for centesimal argu-
ments, viz. from 0' to 10", at intervals of 10", and from 0' to 50', at in-
tervals of 10' to 11 places, without differences (», \ " being used to denote
centesimal degrees (or grades as they are sometimes called), minutes, and
seconds).

[T. IV.] Hyperbolic logarithms of numbers from 1 to 1000 to 11 places.

[T. v.] Log differences of sines for every 1", 2*, . . . 10" throughout the
quadrant, and the same for tangents for 1" and 2', to 7 places, viz. log

sin 2»— log sin 1", log sin S*— log sin 2* throughout the quadrant of

XOO", log sin 4»— log sin 2', log sin 6"— log sin 4* throughout the quadrant,
&c. It is to be noticed, however, that in this mode of description of. the
table log sin 0» must be treated throughout as instead of — oo ; for facing
1' we have given log sin 1» (not log sin V— log sin 0") in the first column;
and facing 2» in the second we have log sin 2" &e.

[T. VI.] A great centesimal table, giving log sines, cosines, tangents, co-
tangents, secants, and cosecants from 0" to 3», at intervals of 10" (with full
proportional parte for every second), thence to 50' at intervals of 1*, with
full proportional parts for every 10").

A page of tables for converting sexagesimals into centesimals &c., com-
pletes the work, which is a thick small-sized quarto, with clearly printed
and not too heavy pages. After the printing of the work Prony asked
Delambre to examine the Tables dtj Cadastee (which are to every 10"
throughout the quadrant to 12 places ; but see § 3, art. 13) ; and this gave
Delambre the opportunity of reading them with Borda's table of sines and

ON MATHEMATICAL TABLKS. 89

tangents in this work : the result was the detection of a great number of
last-place errors, which are given on pp. 117-119 (see p. 114, Preface de
I'editeur). There are other errata given on p. 116.

De Morgan remarks that Delambre is wrong in saying that Hobeet and
Ideleb's tables, 1799 (§ 4), subdivided the quadrant as minutely as those
which he and Borda had published ; but this is not the case, as the latter
are as stated above. The mistake is one into which any one accustomed
to describing tables would naturally fall, as the mode of arrangement gives
the impression, that the portion of [T. VI.] to 3" is to every second, and that
that from 3" to 40" is to every ten seconds : at first sight it is not easy to see
why this was not the form of table adopted ; but the reason for the arrange-
ment being as it is was no doubt that the sine and cosecant, tangent and co-
tangent might be placed exactly on the same footing, as the proportional
parts are the same for each pair. [Mr. Lewis, of Mount Vernon, Ohio, men-
tions that Bremiker has fallen into the same mistake as De Morgan did, thus
giving additional proof of how misleading is the arrangement of the table to
those who have not had occasion to use it : see ' Monthly Notices of the
Eoyal Astronomical Society,' May 1873, vol. xxxiii. pp. 455—458.]

Bowdit'ch, 1802. T. XII. Por the conversion of arc into time.

T. XIII. Log 5 elapsed time, mid time, and rising ; same as T. XVI. of
Maskeitnis, 1802. It is stated in the preface that this table was first
published by Mr. Douwes, of Amsterdam, about 1740, and that he re-
ceived £50 for it from the Commissioners of Longitude in England.
1024 (small) errors contained in this table in the second edition of REaxrisiiE
Tables are said to be here corrected.

T. XIV. Natural sines for every minute to 5 places.

T. XV. Proportional logarithms for every minute to 3° ; same as T. 74 of
Eapee.

T. XVI. Log sines, tangents, and secants for every quarter point to 5
places, and five-figure logarithms to 10,000.

T. XVII. Log sines, tangents, and secants for every minute of the qua-
drant to 5 places : arguments also in time {^Q°=twelve hours), and the com-
plement to 1 2'' given also. The other tables are nautical.

On the titlepage it is stated that the tables are " corrected from many
thousand errors of former publications ; " most of them doubtless only affect-
ing the last figure by a unit.

Bremiker, 1852. T. I. Six-figure logarithms to 1000, and from 10,000
to 100,010, with proportional parts; with degrees, minutes, and seconds
corresponding to every tenth number of seconds, and ten times each such
number; the change in the line is denoted by a bar over the 3rd figure
in aU the logarithms affected. The table is followed by the first hundred
multiples of the modulus -434 ... and its reciprocal to 7 places.

T. II. Log sines (left-hand pages) and tangents (right-hand pages) for
every second to 5° to 6 places, and log sines and tangents for every ten
seconds of the quadrant to 6 places, with differences, and proportional parts
beyond 5°- This is followed by small tables giving the circular m^hsure of
1° 2° . . . 180°, 1', 2', . . . , 60', 1", 2". . . 60" to 6 places ; and for the
conversion of arc into time &c. The last page contains a few constants.

There is an introduction of 82 pp., containing, among other things, an in-
vestigation " De erroribus, quibus oomputationes logarithmicse aificiuntur."

Nine errors in this work are pointed out by Prof. Waokerbarth in the
' Monthly Notices of the Eoyal Astronomical Society " for April 1867.

Bremiker's Vega, 1857. T. I. Seven-figure logarithms to 1000, and

90 KEPORT— 1873.

from 10,000 to 100,000, with differences and all the proportional parts on the
page. The change of figure in the line is denoted by;a fear placed over the
fourth figures of all the logarithms affected. S and T (see § 3, art. 13) are
given at the bottom of the page, as also are the numbers of degrees, minutes,
and seconds corresponding to every tenth number in the number-column of
the table. At the end of this table is a table containing the first hundred
multiples of the modulus '434 . . . and its reciprocal 2-302 ... to 7 places.

T. II. Log sines and tangents from 0° to 5° to every second, to seven
places : no differences. At the end of this table is given a page of circular
arcs, containing the circular measure of 1°, 2°, . . . 180° ; 1', 2', ... 60' ; 1",
2 ', . . . 60" to seven places.

T. III. Log sines and tangents for every ten seconds of the quadrant, to
seven places, with differences : proportional parts are added after 5°-

T. III. is followed by a page containing tables for the conversion of arc
into time : the other tables are astronomical. On p. 547 are a few con-
stants. The tables are stereotyped.

An edition with an English Introduction, edited by Prof. W. L. F.
Fischer, was published in 1857 (title in § 5) ; the contents are the same as
in the above work, the tables being printed from the same plates.

Brolins, 1870. T. I. Seven-figure logarithms of numbers to 1000,. and
from 10,000 to 100,000, with differences, and all the proportional parts.
The all is printed in italics, because in Babbagjs, Callbt, &p. only every other
table of proportional parts near the beginning of the table is given, for want
of space.

In this work there is no inconvenient crowding, as even where the side-tables
are very numerous, the type, though small, is still very clear. The constants
S and T, for the calculation of sines and tangents (§ 3, art. 13), are added,
and placed at the bottom of the page, as also are the numbers of degrees,
minutes, and seconds in every tenth number of the number-column (regarded
as that number of seconds), and the same for each of these numbers multi-
plied by' 10.

T. n. Log sines, cosines, tangents, and cotangents to every second from
0° to 6°, to seven places, with differences throughout, &ndu proportional parU,
except in the portion of the table from 10' to 1° 20', where the size of the
page would not admit of their insertion.

T. in. Log sines, cosines, tangents, and cotangents from 6° to 45° to
every ten seconds, to seven places, with differences and proportional parts.
Of course room could not be found for the proportional parts of all the dif-
ferences ; but throughout all the table on no page are there less than six
proportional-part tables.

On p. 186 the first hundred multiples of the modulus and its reciprocal
are given, to ten places ; and at the end of the book are tables of circular arcs,
viz. the circular measure of 1°, 2°, . . . 180°, 1', 2', . . . 60', 1", 2", . . . 60",
to ten places, a page for the conversion of arc into time, and some constants.
In T. I. the change in the line is denoted by a bar placed over the fourth
figure of all the logarithms affected, the similar change when the third figure
is decreased being denoted in the other tables |)y an asterisk; a final 5 in-
creased has a bar superscript. It is incorrectly stated in the preface that thp
practice of marking all the last figures that have been increased was intro-
duced by ScHBoir ; for this innovation was due to Babba&b (see his preface,
p. x). Dr. Bruhns may, however, merely mean that the mark (viz. a bar sub-
script) introduced by Schbon (1860) fatigues the eye and is of next to no
use; and if so, we entirely agree with him. In Babbagb the increase is

ON MATHEMATICAL TABLES. 91

denoted by a point subscript, which the reader scarcely notices; but in
Schron the bar catches the eye at once and is confusing. The cases also
in which it is necessary to know whether the last figure (unless a 5) has been
increased are excessively rare ; and in fact any one who wants such accuracy
should use a ten-figure table.

On the whole, this is one of the most convenient and complete (considering
the number of proportional-part tables) logarithmic tables for the general com-
puter that we have met with ; the figures have heads and tails ; and the pages
are light and clear. Further, we believe it is pubhshed at a low price.

Byrne, 1.849 (Practical . . . method of calculating &c.). [T. I.] Primes
to 5000, pp. xiii and xiv.

[T. II.] A very small table to convert degrees &o. into circular measure,

p. XV.

[T.' in.] List of constants (69 in number), chiefly relating to tt (which
Mr. Byrne denotes by p), such as 2 tt, 36 tt, -^j it, tt V2, n/n, &c. (pp. xviii
to xxiii) : the value of ir is inaccurate ; see § 3, art. 24.

[T. IV.] Logarithms of numbers from unity to 222, to 50 places (pp. 77-82).

Callet, 1853. [T. I.] Seven-figure logarithms to 1200, and from 10,200
to 108,000 (the last 8000 being to 8 places). Differences and proportional
parts are added ; but near the beginning of the table, where the differences
change very rapidly, only the proportional parts of alternate differences are
given, through want of room on the page (this is also done by Babbagb and
others). The constants S and T (see § 3, art. 13) for calculating the log
sines and tangents of angles less than 3°, as also V the variation for 10",
are given in a line at the top of the page (see p. 113 of the Introduction).
To the left of each number in the number-column are placed not only the
degrees, minutes, &c. corresponding to that number of seconds, but also, in
another column, those corresponding to ten times that number. When the
change of figure occurs in the middle of the block of figures the line is broken
— the best theoretical way of overcoming the difficulty. De Morgan and
others, however, have expressed a strong dislike to it ; and we agree with
them.

[T. II.] I. Common and hyperbolic logarithms of numbers from 1 to 1200
to 20 places, the former being on the left and the latter on the right-hand
pages. II. Common and hyperbolic logarithms of numbers from 101,000 to
101,179 to 20 places, with first, second, and third differences, the hyper-
bolic logarithms being on the right-hand pages. (Note. All the common
logarithms from 101,143 to 101,179, with one exception, contain errors.)
III. Common and hyperbolic antUogarithms from -00001 to -00179 at
intervals of -00001, and from -000001 to -000179 at intervals of -000001,
respectively, to 20 places, with first, second, and third differences.

[T, III.] I. Common logarithms (to 61 places) and hyperbolic logarithms
(to 48 places) of all numbers to 100, and of primes from 100 to 1097 ; and
(II.) from 999,980 to 1,000,021 : the hyperbolic logarithms occupy the right-
hand pages as before.

[T. IV.] The first hundred multiples to 24 places, and the first ten mul-
tiples to 70 places, of the modulus -434 . . . and its reciprocal 2-302 . . .

[T. v.] Eatios of the lengths of degree &c. (ancient and modern) to the
radius as unit, viz. the circular measure of 1°, 2°, ■ ■ ■ 100°, 1', 2', . . . 60',
1", 2", . . . 60", and of the corresponding quantities in the centesimal divi-
sion of the right angle (1' . . . 100'; T . . . 100^; 1". . .100") to 25 places.

[T. VL] Log sines and tangents for minutes (centesimal) throughout the
quadrant (to seven places), viz. frdm 0' to 50", at intervals of 1', with differences.

92 REPORT— 1873.

The order of the eolumjis is sine, tangent, diflferenee for sine, difference for
tangent, cosine ; but this arrangement only holds up to 5", when differences
are added for the cosine also. A change in the figure at the top of the
column is denoted in the column by a line subscript under all the figures of
the firs*^ logarithm affected, which arrests the eye at once.

[T. VII.] Natural and log sines (to 15 places) for every 10' (ten minutes
centesimal) of the quadrant. It is as well here to note that in the log sine
and cosine columns only nine figures are given, as the preceding figures are
obtainable from [T. VI.] ; two, however, are common to both : thus from
[T. VI.] we find log sin 10'=7-1961197, and in [T. VII.] we have given,
coriesponding to log sin 10', 969843372; so that log sin 10'=7-19611969
843372. It will therefore be noticed that the log sines are in strictness
given to' 14 (and not 15) places. Further, it appears that the last figure
has not been, or at all events not been always, corrected; for. log sin 50'=

log A^= -34948500216800940. . . ., and the logarithm in [T. VII.] ends

v2
with the figures 6800. This is the only one we have examined.

At the end of [T. VII.] is given a page of tables to connect decimals of a
right angle with degrees, minutes, and seconds, &c.

[T. VIII.] consists of proportional-part tables, and occupies 10 pp. : by
means of them any number less than 10,000 can be multiplied by a single
digit with great ease ; the use of this in interpolation is evident. A full
explanation is given on pp. 32-36 of the Introduction to the work.

[T. IX.] Log sines and tangents for every second of the first five degrees,
to seven places, without differences (sexagesimal).

[T. X.] Log sines and tangents for every ten seconds of the quadrant, to
seven places, with differences (sexagesimal).

[T. XL] Logistic logarithms, viz. log 3600" - log x" from x = 0" to
.a; = 5280"=l°28'; 3600"=1°.

The other tables have reference to Borda's method for the determination
of the longitude at sea.

On the whole, this is the most complete and practically useful collection
of logarithms for the general computer that has been published. In one not
very thick octavo volume, 11 important tables are given ; the type is very
clear and distinct, though rather small. In the logarithms of numbers an
attempt has been made to give rather too much on the page ; but for general
usefuluess this collection of tables is almost unique.

The introduction, of 118 pp., is the worst portion of the work; it is badly
arranged, confused, and, worst of all, has no index ; so that it is very hard to
find the explanation of any table required, if it is explained at all. On
p. 112 the value of e is given ; but the figures after the 8th group of five

are erroneous, and should be 47093 69995 95749 66967 6 (see Brit.

Assoc, lleport, 1871, Transactions of Sections, p. 16).

On pp. 12 and 13 of the introduction are two tables that deserve notice :
the first gives the square, 4th, 16th .... 2™th roots of 10 to about 28 significant
flwures (leaving out of consideration the ciphers that follow the 1 in the
higher powers). The second gives powers of -5 as far as the 60th.

With regard to errors, an important list is given by Lefort in the ' Comptes
Eendus,' vol. xliv. p. 1100 (1857) ; and these of course apply to the later
tirages. Many errors of importance, as also some information as to the
sources whence CaUet derived his tables, are given. See also Gauss in Zach's
' Monatliche Correspondcnz,' November 1802 (or 'Werke,' t. iii. p. 241), for
four errata, and Gernerth's paper (referred to at the end of the introductory

ON MATHEMATICAL TABLES. 93

remarks in §3, art. 13), and also Humou's tables (editions 1783-1822).
Gernerth remarks (p. 25) that errors pointed out by Button in 1822 still re-
mained imcorrected in the tirage of 1846. "We may also refer to a paper by
Herrmann, entitled " Verbesserung der II. Callet'schen Tafel der gemeinen
Logarithmen mit 20 Decimalen, nobst Vorschlagen fiir die weitere Porde-
rung dieses Zweekes," printed in the ' Sitzungsberichte der Kaiserlichen
Akademie der Wissenschaften,' Vienna, 1848, part ii. pp. 175-190.

On p. liii of their work, Hobekt and Ideleb (1799) remark that they
found that in general the natural sines of Callet were calculated accurately,
but that in the log sines the last two figures were generally doubtful ; they
mention also that they found many other faults in the work, but, being un-
certain how far these are corrected in the stereotype edition, they only give
one : viz., on p. 117 of the introduction, in the eighth place in the value of/
there is a 2 for a 3 ; and this fault renders erroneous the multiples of /. A
list of 380 errors is given on pp. 348 and 349 of the book, in aU of which
the error is + 1 in the last place, and also an error in a natural sine is given.
The above error in / is corrected in the tirage of 1853.

On p. 120 of BoKDA and Dblambke there are given six errors in the ste-
reotyped tables of Callet. A good many errors are also given at the end of
Vega's Manual (1800).

Many other errata are noted in other books ; but it seems useless to give
references without at the same time .examining whether the errors have been
subsequently corrected, and, if so, in what tirages.

Hobert and Ideler consider that Callet obtained his log sines most pro-
bably by interpolation from the ' Trigonometria Artiflcialis ' of Vlacq.

The number of tirages of this work has been very great : it was first
published in 1783, we believe ; but the type from which the earlier tirages
were printed was subsequently reset, as the size of the page in the editions
published in this century is larger than that of the first, which had therefore
more right to the title " Tables portatives." The tirage we have described
above is that of 1853 ; and we have seen one of 1862, " revue par J. Dupuis "
(Dupuis was himself subsequently the editor of a set of logarithmic tables,
described in this section). There is also a stiU. more recent edition, edited
by M. Saigey. We have an impression that the Callet of 1793 was the first
logarithmic table stereotyped ; but we have not investigated the matter.

Coleman, 1846. T. XIX. Log sines, tangents, and secants to every
quarter point, to 6 places.

T. XX. Six-figure logarithms to 10,000, arranged in decades, with pro-
portional parts above 1000.

T. XXI. Logarithms for finding the apparent time or horary angle, viz.

log semi- versed sines ( =log g ) ^"^^"^ ^^ *° ^^ ^* intervals of 5', to

5 places, with proportional parts.

T. XXin. Log sines, tangents, and secants for every minute of the
quadrant, to 6 places.

T. XXIV. Proportional logarithms for every second to 3 ; same as T. 74
of Eapbe, only to 5 instead of 4 places. It must be observed that on the
first page (extending to 10') the logarithms are not given completely, the
last figure, two figures, or three figures being printed as ciphers. This
is done, we presume, because in the oases to which the table is intended to
be applied accuracy in these places is not required. The same is done in
several other copies of this table occurring in other nautical collections.
Opposite is given 4 . 88 . . instead of — oo. The other tables are nautical.

9J! REPORT 1873.

Croswell, 1791. T.I. Logsecant8,halflogseeaiitB,andhalflogsines,viz.
log sec X, I log sec a; and | log sin x, to every minute! of the quadrant, to seven
places, the last two being separated by a comma for the convenience of those
■who only requii-e five places ; semiquadrafltaUy arranged : no differences. The
table, as headed in the book, impUes that the tabular results are natural ;
but they are as above.

T. V. Proportional logarithms for every second to 3°, to 4 places: the
same as T. 74 of Eapek.

T. XTTI. Small table to convert arc into time. The other tables ai'e
nautical.

De Decker, 1626. T. I. Ten-figure logarithms of numbers to 10,000,
with characteristics and differences.

T. II. Logarithmic sines and tangents, to seven decimals, for every minute,
from Qttntee 1620 (§ 3, art. 1,5).

These tables were always assigned to Vlacq till, in the course of the pre-
paration of this Eeport, it came to light that De Decker was the author, Vlacq
having only rendered some assistance. For the history of them, as well as
for their connexion with ' Tables des Logarithmes pour les nombres d'un k
10,000 composes par Henry Brigge,' Gouda, 1626, and the tables in "Wells's
' Sciographia,' 1635, see PhU. Mag., October and December (Supp. No.), 1872,
and May, 1873.

Degen, 1824. T.I. Logj„(1.2.3 a:)isgivenfroma!=l toa;=]200,

to 18 places. The complement of the logarithms from 100 is also added if the
characteristic be less than 100 — if not, the complement from 1000 or 10,000 ;

thus log (1 . 2 69)=98--233 and the complement is 1'766 ; log

(1.2 70) =100-078 , and the complement is 899-921 The first

portion of this table is reprinted by De Mokqait, to 6 places, in the ' Ency-
clopaedia Metropolitana ' (§ 3, art. 25).

T. II. The first hundred multiples of the modulus -434 . . . , to 30 places.

T. III. The first nine multiples of log 2, log 3, log 5, log 6, log 7^ log 11,
log 12, log 13, log 14, log 15, log 17, log 18, log 19, log 21, log 22, log 23, log 24,
log 26, log 28, and log 29 (Briggian).

The other tables consist of formulae &c. There is a full introduction.

[De Morgan] 1839. [T. I.] Five-figure logarithms to 10,000 (arranged
consecutively, and not as in seven-figure tables), with differences, and degrees
corresponding to the first number in each column.

[T. II.] Logarithms from 1001 to 1100, to 7 places.

[T. III.] Log sines, cosines, tangents, and cotangents to every minute, to
5 places, with differences. i

[T. IV.] Log sines for every second of the first nine minutes, and also for
every tenth of a minute in the first degree.

[T. v.] A small table of constants ; most of them taken from Babbaoe.

[T. VI.] Log (1.2.3. .. .a;), from a!=6. to a7==25, at. intervals of unity,
and thence to 265, at intervals of 5, these last three tables being also to 5
places.

The tables are beautifully priuted, and are practically free from error.
Prof. Wackerbarth. states ('Monthly Notices of the Eoyal Astronomical
Society,' AprU. 1867) that he finds the only error in the work to be among
the constants on p. 213, line 5, where 2-718281829 should be 2-718281828,
the following figure being 4.

There is no name on the titlepage ; but it. is well known that the tables
•were prepared by De Morgan, and they are always spoken of by his name.
They were examined by Mr. Farley of the Nautical-Almanac Office.

ON MATHEMATICAL TABLES. 95

De Prasse, 1814. [T. I.] Kve-figure logarithms of numbers to 339
(with characteristics), and thence to 10,000, arranged as is usual in seven-
figure tables. When the fifth figure has been increased it is printed in difi'erent
type. The change in the line is denoted by an asterisk prefixed* to the third
figure of aE the logarithms aifected.

[T. II.] Log sines and tangents for every minute to 5°, and thence for every
ten minutes to 85°, vrhen the intervals are again one minute to 90°, to 5
places. It and e, and nine multiples of the modulus and its reciprocal are
given on the last page. The price is one franc.

A short review of this work, reprinted from the ' Gdttingische gelehrte
Anzeigen,' Dec. 19, 1814, wiU be found on p. 248 of t. iii. of Gauss's
' Werke.' On pp. 241-243 is also reprinted a review of. the original edition
(Leipzig), from the same 'Anzeigen' for May 25, 1811.

Dodsou, 1747. T. XVII. Least divisors of numbers to 10,000 (mul-
tiples of 2 and 5 omitted).

T. XVni. Primes from 10,000 to 15,000.

T. XIX. Square and cube roots (to 6 places) of numbers to 180.

T. XX. Combinations up to the combination of 34 things, 29 together :
a table of double entry.

T. XXI. Powers of 2 to 2=» &e.

T. XXIi; The first 20 powers of the 9 digits.

T. XXIII. Permutations, viz. 1 . 2. . . .a?, to a;=30.

T. XXV. Circular measure of 1°, 2°,. . . . 180°; of V, 2',.. . .60'; of 1"
.... 60" ; and of 1'" .... 60'"-: to 7 places.

T. XXVI. Versed sines of ares, and the areas of the segments included
by those arcs and their chords to every 15' of the quadrant, to 7 places, with
differences.

1 IT 1

T. XXVII. The first 9 multiples of 12 constants (viz. t,-,-' -r-, &c.),

to 7 places.

T. XXVin. Table of polygons, giving any three of the four quantities,
length of side, radius of inscribed circle, radius of circumscribed circle, area,
when the fourth is given=l, for polygons of less than 13 sides, to 7 places.

T. XXIX. Table of regular solids, giving any four of the five quantities,
side, radius of circumscribed sphere, radius of inscribed sphere, superficies,
solidity, when the fifth is given=l, to 7 places; for the 5 regular solids.

T. XXXII. Seven-figure logarithms to 10,000, with differences.

T. XXXIII. Antilogarithms, viz. numbers to logarithms from '0001 to
•9999 at intervals of -0001, to 7 places.

T. XXXIV. Log sines and tangents for every minute of the quadrant, to
7 places, with diflferences ; but between 0° and 2° the differences between the
■ logarithms of the arcs and the logarithms of the sines and tangents of those
arcs are given instead.

T. XXXV. The number of seconds contained in any number of minutes

less than 2°.

T. XXXVI. Logistic logarithms, viz. log 3600'— log x from cc=l to
a; =4800' (=80"°) (argument expressed in minutes and seconds), to 4 places.

T. XXXVII. Neper's logarithms. The table, however, is really one to con-
vert' common into hyperbolic logarithms, and is in fact, when so regarded, the
first 1000 multiples of the reciprocal of the modulus, viz. 2-302 . . . , to 6 places.

T. XXXVIII. Products to 9 X 9999.

There are besides, very many other tables of all kinds, astronomical, com-
mercial, &c.': we have described all the mathematical ones.

96 REPORT— 1873.

Domke, 1852. T. XXX. Quadrate der Minuten des StundmwinMs, viz.
(« + ^j from a; =1 to at==15, and from j/=l to y=60, to one decimal

place; thus corresponding to 8' 20" the table has 69-4; for 8' 20" = 8^ =
8-33 . . . , and its square, retaining one decimal place, is 69"4.

T. XXXn. Six-6gure logarithms to 100, and from 1000 to 10,000, with
differences : all the logarithms written at fuU length.

T. XXXin. Log sines, tangents, and secants to every quarter point, to
6 places.

T. XXXTV. Log sines and tangents for every second, for the first two
degrees, to 6 places : all the logarithms written at length.

T. XXXV. Log sines, tangents, and secants, to every minute of the
quadrant (arguments also expressed in time), with differences, arranged semi-
quadrantally : all the logarithms written at length.

T. XXXVI. Natural sines to every minute of the quadrant, to 6 places,
arranged quadran tally.

T. XXXVII. Logarithmen der halbverjlossendm Zeit, viz. log cosec a; from
.r=0'' to a; =3'' 59™ 55' at intervals of 5', to 5 places, with proportional parts
for seconds.

T. XXXVIII. Logarithmen der Mitteheit, viz. log 2 sin x, from a;=0*^
to a;=z3^ 59" 55° at intervals of 5°, to 5 places, with proportional parts for
seconds.

T. XXXIX. Logarithmen des Stundenwinkels, viz. log versed sine x, from
x=0^ to x=T^ 59" 55' at intervals of 5", to 5 places, with proportional parts
for seconds.

T. XL. Proportional logarithms for every second to 3°, to 4 places ; the
same as T. 74 of Kapek.

T. XLVII. and XL VIII. occupy one page, and are for the conversion of
arc into time, and vice versd.

The other tables are nautical.

In all the tables the logarithms are written at fuU length ; the type is thin
and very clear, the figures having heads and taUs.

T. XXX. was calculated from this work; T. XXXTL, XXXIIL, and
XXXV.-XL. were taken from Nobie's ' Epitome of Navigation,' (they are
Maskblynb's tables ; but see Bowdiioh, 1802, T. XIII.) and T. XXXIV.
from Cailet.

On the accuracy of this work see the tract of Gernerth's referred to in
§ 3, art. 13 (p. 55). There was a second edition in 1855 (Gernerth).
■ Doun, 1789. T. I. Seven-figure logarithms to 10,000, with differences.

T. II. Log sines and cosecants to every quarter point, to 7 places.

T. III. Log sines and tangents and natural sines for every minute of the
quadrant, to 7 places.

T. IV. Log I elap. time, mid time, and rising (see explanation of the
terms under T. XVI. of Maskeltnb, 1802), for every half minute to 6\ to ■
5 places.

T. V. Log versed sines and natural tangents and secants for every 10' of
the quadrant, to 4 places.

The other tables are nautical.

We have also ' The British Mariner's Assistant, containing forty Tables . . '
London, 1774, 8vo (352 pp. of tables), the tables of which are the same as
those described above.

Douglas, 1809. [T. I.] and T. I. Supplement, and T. II. Supplement,
Logarithms of numbers to 10,999, and from 100,000 to 101,009, to 7 places
(without differences).

ON MATHEMATICAL TABLES. 97

[T. II.] Log sines, tangents, and'socants for every minute of tlio quadranf,
to 7 places (without difi'erenoes).

[T. III.] Natural siues, tajigents, and secants for CTcrj' minute of the
quadrant, to 7 places (without differences).

[T. IV.] Natural and log versed siues to every minute, from 0° to 180°, to
7 places (without differences).

T. III. Supplement. Table to convert sexagesimals into decimals. It
gives 1", 2", 4" . . . 58", 1', 1' 1", 1' 2", 1' 4". . . 1' 58", 2' ... 2' 58", &c. to
60', expressed as decimals of 60', to 4 places.

T. IV. Supplement. Logarithms of numbers from 1 to 180, to 15 places.

Diicom, 1820. T. VII. Proportional logarithms for every second to 3°,
to 4 places ; same as T. 74 of Eapbe.

T. IX. Log sines and tangents for every second to 2° ; then follow log
cosines and cotangents for every 10" to 2°; and then log sines, cosines,
tangents, and cotangents from 2^ to 4-5°, at intervals of 10'', to 6 places.
Proportional parts are added for the portion where the intervals are 10".

T. XIX. Natural sines for every minute of the quadrant, to 6 places.

T. XX. Parties proportionnelles for interpolating when the tabular result

is given for intervals of 24*, viz. ^"^ (expressed in hours, minutes, and

seconds), where a; is 1", 2"", .... 60"", and, in the first table, y is 1", 2", . . . .
24^ and in the second l", 2°-, 60">.

T. XXI. Six-figure logarithms of numbers to 10,800, with corresponding
minutes and seconds : logarithms printed at fuU length ; no diflferences.
The other tables are nautical &c.

The tables form the second part of the work. It may be noticed that, in
the remarks on T. XIX. (p. xiv), the versed sine of x is erroneously defined
as if it were 1— siu x,

Dunn, 1784. [T. I.] Six-figure logarithms to 10,000. The arrangement
is the same as is usual in seven-figure tables j only instead of the numbers
0, 1, 2, .... 9 running along the top line, they are printed 0-00, 1-00, 2-00, ....
9"00, which gives the table the appearance of being arranged differently.

[T. II.] Log sines, tangents, and secants to every minute of the quadrant,
to 6 places. At the foot of each page is a small table, giving the differences
(for the sine and tangent) for an interval of 60" in the middle of the page,
and their proportional parts for 50", 40 ", 30", 20", 10", 9", 8", 7", 6", 5", 4",
3", 2", 1". At the end is a table of the differences of the log sines, tangents,
and secants for every 10'.

Dupuis, 1868. T. I. & 11. Seven-figure logarithms from 1 to 1000, and
from 10,000 to 100,000. Proportional parts to tenths, viz. multiples with
the last figure separated by a comma, are added. (The separation of the last
figure is an improvement on the simple multiples given in Sano, 1871, and
others, as the table can be more readily used by those accustomed only to
joroportional parts true to the nearest unit.) S and T (§ 3, art. 13) are given
at the bottom of the pages at intervals of 10". Dupuis states in the preface
that his intention had been that the table should extend to 120,000, and
that accordingly he had calculated the last 12,000 logarithms by difierences,
but at the request of a number of professors he stopped at 100,000. Wo
venture to think he would have acted more wisely if he had not listened to
the professors*; but the matter is of no consequence now, as 8an», 1871,
extends to 200,000,

* Several of the ordinary seven-flgure tables (Eabeaqe, Callet, HUlsse's Vega, and
many others) extend to 108,000, and the last 8000 logarithms are given to eight places.-

1873. =

98 EEPORT— 1873.

T. Ill, Hyperbolic logarithms to 1000, to 7 places.

T. IV. & V. First hundred multiples of the modulus and its reciprocal, to
7 places.

T. VI. & VII. Log sines and tangents for every second to 5°, to 7 places,
■with negative characteristics (viz. 10 not added).

T. VIII. Log.sines, tangents, cotangents, and cosines (arranged in this
order) from 0° to 45° at intervals of 10", with negative characteristics,
to 7 places ; with differences and proportional parts, as before, to tenths.

T. IX. Circular measure of 1°, 2°, . . . , 180°, 1' . . . . 60', 1" . . . . 60", to 7
places.

T. X. (reduction des parties de I'equateur en temps) ; hours and minutes

(or minutes and seconds) of time in 1°, 2°, 360° (or 1'. .. . 360'.), and

seconds of time in 1", 2", .... 60", to 7 places } then follows an explanation
of the use of the tables.

This is the only work we can call to mind in which negative characteristics
(with the — sign printed over the figure) are given throughout ; and to the
mathematical computer such are preferable to ■ the ordinary characteristics
increased by 10. Also the edges of the pages of T. VI.-VIII« are red (the
rest being grey), which facilitates the use of the tables. It is curious that
it never should have occurred to any editor or publisher of a collection of tables
to colour the edges of the pages of the separate tables differently, and print
thereon also their titles, as is done with the different businesses &c. in the
London Post-Office Directory.

Dupuis was also the editor of the 1862 edition of Cahbi ; and the titles of
several small tables of logarithms that we have not seen are advertised in
this work, viz. : — (1) an editioli of Lalande's five-figure tables, with Gaussian
logarithms added, &C. ; (2) an ISmo book of four-figure tabtes ; and (3)
logarithmic and antilogarithmic tables to 4 places, for the use of physicists,
giving log (l+at) for ttie calculation of dilatations &c.

[Extcke, 1828.] [T. I.] Four-figure logarithms to 100 (with characteris-
tics and differences), and from 100 to 1009.

[T. n.] Log sines, tangents, cotangents, and cosines for every 4' from.
0° to 10°, and thence to 45° at intervals of 10', to 4 places, with dif-
ferences.

[T. III.] Gaussian logarithms j B and are to 4 places, for argument
A, from A="00 to 1'80 at intervals of -01, and thence to 4-0 at intervals of -1,
with differences.

Encke's name is written on the Eoyal Society's copy of these tables ; and
they are also spoken of as Encke's by De Morgan, They are reprinted in

"WaeNSTOEFP's ScHtTMACHEE, 1845 (§ 4).

Everett [1866]. Two cards (one of which, unfolded,is equal in size to three
folio pages, the other, which is equal in size to one, being perforated), in a cover.

This very frequently gives rise to errors, as the computer who is accustomed to three
leading figures common to the block of figures is liable to fail to notice that in this part
of the table there are four; and on this account a figure (the fourth) is sometimes
omitted in taking out the logarithm. It is therefore often desirable to ignore the con-
tinuation of the table and only use the portion below 100,000. The extra logarithms
are thus not always an advantage ; and it is on the face of it inconvenient that some of the
tabular results should be given to 7 and others to 8 places. When tables of logarithms
are placed in the hands of common computers, it is as a rule better to forbid the use of
the portion beyond 100,000 ; and it may have been some considerations of this nature
that induced M. Dupuis to take this number as his limit. But there is no objection that
we can see against giving the logarithms beyond 100,000 to 7 places (as in Sang, 1871) ;
and whenever this is done, the continuation is found very usefuU

ON MATHEMATICAL TABLES. 99

These cards correspond to the fixed and movable portions of a slide-rule
160 inches long. A few small tables of cube roots, sines, &c. are printed on
one of the cards. Prof. Everett (to whom we applied for information with re-
gard to the date of the table) gives the following brief description — " Two
cards, one of them cut like a grating, equivalent to the two pieces of a slide-
rule;" and adds "that in the first edition [which is the one we have
described] one of the cards had a pair of folding leaves attached to it,
but these merely contained subsidiary tables and directions, and were quite
unessential. In the next impression the two essential cards and the two
cards with subsidiary tables and directions were aU detached from each
other." A description of the table is given in the Phil, Mag. for November
1866.

Farley, 1840. [T. I.] Six-figure logarithms to 10,000 (the line is
broken when the change occurs in the third figure) ; followed by the loga-
rithms of numbers from 1001 to 1200, to 7 places.

[T. II.] Log sines and tangents for every minute of the quadrant, to 6
places, with dififerences for 100".

[T. III.] Log sines from 0° to 2° at intervals of 6".

There are also a few constants and some formulae.

Farley, 1856. This very fine table of versed sines contains : — [T. I.]
Natural versed sines from 0° to 125° at intervals of 10", to 7 places, with
proportional parts throughout.

[T. IL] Log versed sines from 0° to 135° at intervals of 15", to 7 places,
with differences throughout. The arguments are also given in time, the
range being from 0* to Q"* to every second.

A short preface by Mr. Hind states that the table was prepared by Mr.
Parley, of the Nautical-Almanac Ofiice, in 1831, and the manuscript pre-
sented by him to Lieut. Stratford, the then superintendent. The manuscript
having been in "use for 25 years, and having become dilapidated, it was
" deemed the most economical course to print it." It is added that the last
figure cannot be relied on, though it is probably very rarely in error by more
than a unit.

These, the most complete tables of versed sines we have seen, are beauti-
fully printed, in the same type as the Nautical Almanac.

Faulhaber, 1630 (' Ingenieurs-Schul '). The copy we have seen of this
book (viz. that in the British Museum) contains no logarithms, though it must
evidently have been intended to accompany some tables. In the Brit-Mus.
copy the work is bound up (in a volume containing four tracts) after the two
described below and attributed by us to Faulhaber. Murhard gives the
full titles of this work and of the next two, and marks them as having come
under his eye ; he does not, however, assign the two tables to Paulhaber,
Eogg, who also gives the titles of the three works, attributes them all to Paul-
haber. He adds, speaking of the tables, that they are also contained in the
' Ingenieurs-Schul.' This is no doubt correct; for, as noted below, some errors
in the latter work are given at the end of the Canon. It seems therefore^
certain that Faulhaber was the editor of the tables. It may be mentioned
that both Eogg and Murhard agree in describing the 'Logarithmi' and the
' Canon ' as parts of the same work, so that most likely they were never issued
separately. Eogg gives the date of the ' Ingenieurs-Schul' as 1731, which
must be a misprint for 1631 ; the copy before us is dated 1630, agree-
ing with Murhard. A lengthy account of Faulhaber and his works will
be found in Kastner's ' Geschichte.' Sec also.Soheibel, ' Math. Biicherk.' B. 2.
P. 39.

100 heport— 18/3.

[Patdhaber] 1631 ('Logarithmi'). Seven-figure logarithms of numbers
from 1 to 10,000, arranged in columns (three to the page), with charac-
teristics. As there arc 3 columns, there are 99 logarithms on each page. The
printing is imperfect, the types having here and there become displaced,
so as to leave no mark. There are some errata on the last page, headed
" Typographus Lectori S." See above, Eattlhabee, 1630 (' Ingenieurs-
Schul').

[Faulhaber] 1631 (-'Canon'). Logarithmic sines, tangents, and secants
for every minute of the quadrant, to 10 places (semiquadrantally arranged) ;
no differences. Taken from Vlaco, 1628. The table is followed by 8 pages of
errataintheFrankfort ' Ingenieurs-Schul,' in the logarithmsof numbers, andin
the ' Canon.' Except perhaps Norwood, 1631, this is the first reprint of
VLAca's corrected ' Canon ' (1628), the previous writers having copied
GiTNTEE (1620). Kogg gives place and date as Nuremberg, 1637 ; but
the copy before us is not so. See above, Faitlhaber, 1630 (' Ingenieurs-
Schul').

Filipowski, 1849. T. I. AntUogarithms. The numbers (to 7 figures)
are given answering to the logarithms as arguments, the range being from
•00000 to 1-00000 at intervals of -00001. The arrangement is exactly the
same as in ordinary seven-figure tables of logarithms ; and the table occupies
201 pages. The proportional parts are given to hundredths (viz. 100 pro-
portional parts of each difference are given); and the change of figure in the
middle of the line is denoted by two dots (thus, 0) placed over the fourth
figure of aU numbers affected ; and when a final 5 has been increased it is
printed V. The first 3 figures in the number are alwaj's separated by a
space from the block of figures.

T. II. Gau.ssian logarithms, arranged in a new way. Let A=log co and
X=log(.r+l)(sothatlO*^=10'^+l),then on the first page of the table (p. 203
of the book) we have A given to 3 places for argument X from \= -00000 to
•00449 (which last corresponds to A=8-017), at intervals of -00001. On
the succeeding 16 pages we have X as a tabular result for argument A from
A= 8-000 to 13-999, at intervals of -001, to 5 places.

Since log (a+6)=log 6 -i-log (^ + l), and

log (a-5)=log 5+log (^-l).

it is clear that the rules are very simple and uniform, viz. log a and log 5
being given (6 <cj suppose), we take log a- log 6 as argument, and enter
the table at the A or X column, according as we want log a+h or log a—h,
and add the tabular result to log 6. In this table also the notations 0,
V, &c. are used, as well as another in which a wavy line runs down by the,
side of the logarithms whose leading figures have changed. This method of
marking is only possible when the tabular results appear one under the other.
The figures are throughout neat and clear, having heads and tails ; and the
copy before us is printed on green paper, of a pleasant colour. In many
places there is a parsimony of figures, which we dislike extremely ; thus there
occur 44, 5, 6 as headings for 44, 45, 46, and or 6 for 10 &c. A list of 36
errors affecting the first 8 figures of Dodson's Canon (1742) is given, and in-
troduced by the remark, " The following is a list of errors as detected, by
means of our table, in the first 8 places of Dodson's Anti-Logarithmic Canon,
in addition to those corrected with the author's own hand." These words im-

ON iMATHBMATICAL TABLES. 101

ply that Mr. Filipowski's table was the result of an independent calculation ; or
at all events they ought not to have been written unless such had been the case.
It is, however, nowhere stated in the preface that the table was calculated
anew ; and we may therefore assume that it was copied from Dodson, after
examination (which would not have- been difficult, as a mere verifloatiou by
differences would have sufficed). In a letter by Mr. Peter Gray, in the
' Insurance Record' for June 9, 1871, there are given two errors in Dodson
which also occur in FiUpowski, affording additional evidence that the tables of
the latter were not calculated independently ; and, this being so, Dodson
has not been treated fairly, as Mr. Pilipowski should have acknowledged the
obligations he was under to his table. In the same letter Mr. Gray
gives three other errors in Pihpowski (1st edit.) ; and it is to he in-
ferred from other passages in the letter that a second and a third edition,
" corrected," have been published. Mr. Gray proceeds :— " but he [Fili-
powski] has never, so far as I know, given a list of the errors contained in the
first and second, and corrected in the third," an omission on which he strongly
(and most justly) animadverts. See Shobtbede (1849).

De Morgan has stated that no antilogarithmic table was published from
Dodson (1742) till 1849 ; but this is only true if Shobtkede's tables of 1844
be ignored ; for which there is no sufficient reason, as they were published
and sold in that year, and copies of the 1844 edition are contained in all good
libraries.

Galbraith, 1827. T. II. Six-figure logarithms of numbers to 10,000,
with proportional parts on the left-hand side of the page. This table is
headed " Logarithms of numbers to 100,000."

T. IV. Log sines, tangents, and secants to every quarter point, to 6 places.

T. V. Log sines, tangents, and secants to every minute of the quadrant
(arguments expressed also in time, the intervals being 4'), with differences,
to 6 places.

T. VI. Natural sines, tangents, secants, and versed sines to every degree
of the quadrant, to 6 places.

T. IX. Diurnal logarithms : proportional logarithms for every minute
to 24'' (viz. log 1440— log «) from 00=1 to .r=1440 (expressed in hours and
minutes), to 5 places.

T. X. Proportional logarithms for every second to 3°, to 5 places. Same
as T. 74 of Eapeb, except that 5 instead of 4 places are given.

T. LXIII. A few constants. The other tables are nautical.

There are a few small tables in the introduction that may be noticed, viz. : —
T. XI. and XII. (p. 113), to express hours as decimals of a day, convert
lime into arc, &c. ; T. XV. (p. 141), of the areas of circular segments
(same as in T. XIII. of Hantschi, but to hundredths only, and to 5 places) ;
and T. XVI., table of polygons (as far as a dodecagon), giving area, and radius
of circumscribing circle for side=unity, andfactors for sides, viz. length of side
for radius = unity ; there are also one or two small tables for the mensuration
of solids.

Galbraith and Haughton, 1860. [T. I.] Five-figure logarithms to
1000, arranged in columns. This is followed by a small table to convert
common into hyperbolic logarithms, and vice.versd.

[T. II.] Five-figure logarithms from 1000 to 10,000, with proportional
parts.

[T. III.] Log sines and tangents to every minute of the quadrant, to 5
places, with differences.

[T. IV.] Gaussian logarithms. B and C are given for argument A, from

102 KErouT-1873.

A=-000 to A=2'000 at intervals of -001, theuco to 3-40 at intervals of -01
and to 5 at intervals of -1 to 5 places, •with differences. This table is followed
by a page of constants,

Gardiner, 1742. [T. I.] Seven-figure logarithms to 1000, and from
10,000 to 100,100, with proportional parts ; the change of the fourth figure
in the line is not marked ; the first three figures of the logarithm are sepa-
rated from the block of figui'es by a point, which is very clear.

[T. II.] Log sines to every second to V 12", to 7 places, without diffor-
enees; and log sines and tangents throughout the quadrant at intervals of 10",
to 7 places, with differences.

[T. III.] Four-figure logistic logarithms, viis. log. 3600"— log tv from te—O
to a;=4800" (=80') at intervals of 1".

[T. IV.] Twenty-figure logarithms to 1000, thence of odd numbers to
1069, and of primes &e. to 1143.

[T. v.] Twenty-figure logarithms of numbers from 101,000 to 101,139,
with first, second, and third differences.

[T. VI.] Anti-logarithms, viz. numbers to logarithms from -00000 to
•00139 at intervals of -00001, to 20 places, with first, second, and third dif-
ferences.

A list of errata is given in the French reprint described below ; and 69
errors are pointed out by Hutton on p. 342 of the edition of 1794 (and
no doubt in other editions) of his mathematical tables. The list given in the
edition of 1822 (the last published in Button's lifetime) is much fuUer, Do
Morgan speaks of Gardiner as "rare, and much esteemed for accuracy;" and
its rarity in 1770 is the reason assigned by the French editors for the neces-
sity of reprinting it.

Gardiner (Avignon Eeprint, 1770). The reprint is so similar to the ori-
ginal edition thatoit is only necessary to point out the differences,

[T. I.] is the same ; but in [T. II.] the log sines are given at intervals of
1" as far as 4°, and a similar table of log tangents is added ; they were taken
from a manuscript calculated by Mouton, bequeathed by him tp the Academy
of Sciences, and lent to the editors by Lalande. Also in the original edition,
in the second portion of this table, viz. that giving the functions at intervals
of 10", the parts common to both are repeated ; but this is not done in the
reprint, in which therefore there is a table of log cosines and cotangents only,
from 0° to 4°, at intervals of 10", the sines and tangents being given in the
previous portion.

[T. III., v., and VI.] are unaltered ; but [T. IV.] proceeds by odd numbers
to 1161. One fresh table is added, viz. [T. VII.], giving hyperbolic loga-
rithms from 1-00 to 10-00 at intervals of -01, to 7place8, and also log, 10^, , . . 10^
Mouton's manuscript also gave log cotangents and cosines to every second
of the first four degrees ; but the former are so easily deducible from the tan-
gents, and the latter vary so slowly, that their publication in ewtenso seemed im-
necessary. A page of errata at the end of the book contains errors in Vlacs
(1628), in GAEmiTEB (1742), and in the French reprint itself (1770), the last
having been published in the ' Connaissance des Temps ' for 1775. As the
' Connaissance des Temps' could not have been published as much as five
years in advance, it is clear either that some copies of the French reprint were
published subsequently to 1770, although retaining thatdate on the titlepage,
or that this page was circulated separately and bound up afterwards with the
work. "We have examined two copies, in one only of which this errata-page
appears.

No editors' names appear ia the work ; but Lalande (Bibliog.Astron. p. 516)

ON MATHEMATICAL TABLES. 103

says that this edition was edited by P^re Pezenas, P&re Dumas, and Pere
Blanchard, and adds that he has given an errata-list in the ' Connaissance
des Temps ' for 1775. On Dumas, mathematioian of Lyons, who was La-
lande's &st master, he gives a reference to the ' Journal de^ Savants,' No-
vember 1770.

The edition is very commonly known by the name of Pezenas. A good
deal about Pezenas will be found in Delambre's ' Histoire de I'Astronomie,'
pp. 368-386. He was born at Avignon in 1692, and died in 1776.

The French edition is even better printed than the original, but is not
quite so accurate. A list of 85 errors is given by Hutton on p. 343 of bis
mathematical tables in the edition of 1794, while he discovered only 69
in the original edition; more complete lists are to be found in the later
editions.

Graesse (' Tresor') gays that there was a reprint of (Jardiner in octavo at
ITlorence by Canovai and Kicco.

^Gardiner (Paris edition, 1773), Eogg gives the title of a Paris edition
of Gardiner, viz. ' Tables des Logarithmes de Gardiner, fol., Par. Chez Sail-
lard et Nyon, 1773,' which he takes from the • Journal litteraire do Berlin,'
t. vii. p. 318 ; but the fact that Lalande does not mention it seems to him
very suspicious : we have seen no other reference to it, and agree with Rogg.

Qarrard, 1789. This work contains only traverse and meridional part
tables. It is referred to here, as its title would imply that it was included
in the subject of the Report.

Gordon, 1849. T. IX. Log sines, tangents, and cosecants for every
minute from 6° to 90°, to 4 places.

T. X. Proportional logarithms for every second to 3°, to 4 places : same
as T. 74 of Eaper.

T. XL Small table to convert space into time.

T. XVIIi Half-sines and half-dosines, vizf. halves of natural sines for
every minute of the quadrant to four places, reckoned as seconds for the,
purpose of adaoting them to the table of proportional logarithms : thus, cor-
responding to 1^° 40' we find as tabular result 18' 16" ; for the number of
seconds in this angle=1096, and | sin 12° 40'=-1096 ...

T. XVIII. Logarithms of the meridian distance, viz. log (^ vers sin x),
from ^7=0'' to a;=7^ 59"" 55" at intervals of 5^ to 4 places.

T. XIX. Proportional logarithms for every minute to 24% viz. log 1440
—log X from (v=l to a7=1440, to 4 plaqes (arguments expressed in hours
and minutes).

T. XXI. Proportional logarithms for one hour, viz. log 3600— log a;
from x=l to a;=3600, to 4 places (arguments expressed in minutes and
seconds).

The other tables are nautical.

Gregory, Woolhouse, and Hann, 1843. T, VIII, Proportional
logarithms for every second to 3°, to 4 places ; same as T. 74 of Eapee.

T. IX. Log sines, tangents, and secants for every minute of the quadrant,
to 5 places.

T. X. Natural sines to every minute of the quadrant, to 5 places.

T. XI. Pive-figure logarithms from 1000 to 10,000, with proportional
parts.

T. XII. Proportional logarithms for every minute to 24", to 4 places, viz.
log 1440-^log« fromi»=al tp,1440 at intervals of unity (arguments ex-
pressed in hours and minutes).

The ptJier tables are nautical.

104 REPORT — 1873.

Griffin, 1843. T. 16. Log sines, tangents, and Becants to every quarter
point, to 6 places.

T. 17. Six-figure logarithms of numbers to 100, and from 1000 to 10,000,
to 6 places, -with differences.

T. 18. Log sines, tangents, and secants to every minute of the quadrant
(arguments expressed also in time), to 6 places, with differences for the sines
and tangents ; arranged semiquadrantally.

T. 19. Natural sines to every minute of the quadrant, to 6 places,
without differences.

T. 41. Proportional logarithms to every second to 3°, to 4 places; same as
T. 74 of Eapeb.

The logarithms are in all the tables printed at full length. The other
tables are nautical.

Gnison, 1832. T. I. Seven-figure logarithms to 10,000 : no differences.
The change in the line is marked by a difference of type in all the logarithms
affected. In three or four parts of the book this table is stated to extend to
10,100, but the limit is as above ; and there is no possibility of a page having
been torn out, as the next table is printed on the back of the page ending
with the number 9999.

T. II. & III. Squares and cubes of all numbers from 1 to 1000.

T. IV. & V. Square and cube roots of all numbers from 1 to 1000, to 7
places.

T. VI. Circular measure of 1°, 2°, 3° . . . 360°, of 1', 2', . . . 60', and of
1", 2", . . . 60", to 7 places.

T. VII. Natural and log sines, cosines, tangents, cotangents, secants, and
cosecants, to 7 places, with differences from 0° to 5° at intervals of 1', and
thence to 45° at intervals of 10'.

The book was intended for schools.

Hantschl, 1827. T. I.- Five-figure logarithms (written at full length)
of numbers from 1000 to 10,t00.

T. II. Log sines for every 10 seconds from 0° to 90°, to 6 places. ■

T. III. Log tangents for every 10 seconds from 0° to 90°, to 6 places.

T. IV. Ten-figure logarithms of primes to 15,391.

T. V. Natural sines, tangents, secants, and versed sines for every minute
of the quadrant, to 7 places ; arranged semiquadrantally.

T. VI. Hyperbolic logarithms of numbers to 11,273, to 8 places.

T. VII. Least divisors of numbers to 18,277 (multiples of 2, 3, 5, and
11 excluded).

T. VIII. Squares, cubes, square and cube roots (to 7 places) to 1200,

m TY n(n—l) n(n—l) . . . («— 5) - „ , t nn ,

T. IX. ^ -' , . . . -i — ,r-^ *■ •' from n=0 to w=l-00 at

intervals of -01, to 7 places.

T. X. Circular measure of 1"; 2°, 3°, ... 180°, of 1', 2' . . . 60', and of
1", 2" ... 60", to 15 places.

T. XI. The first nine multiples of

1 «■ ?r
"' ^' 4' 6' S' 12'

to 24 or 21 places.

T. XII. Small table to express minutes and seconds as decimals of a
degree.

T. XIII. Areas of segments of circles for diameter unity to 6 places : the

i. fi. # (ly. - (i)-

ON MATHEMATICAL TABLES. 105

! versed sines are the arguments ; and the table proceeds from -001 to -500 (of
Ithe diameter). The table may therefore be described as giving J(29— sin 20)
'from 1(1— cose) = -001 to -500 at intervals of -001.

A few constants are then given to a great many places ; and the last page
(T. XIV,) is for the calculation of logarithms to 20 places.

The -work is clearly printed.

Hartig, 1829. The tables 'are of so commercial a kind that only one or
two deserve notice here.

The first (T. I.) is for computing the contents of planks &c., the thickness and
breadth being given in Zolle and the length in Fusse, and may be described
as a sort of duodecimal table, as the Kubik-ZoU = ^^ Kubik-Euss, and the
Kubik-Linie = Jj- Kubik-Zoll. Thus for arguments 3 ZoU, 13 ZoU, and
5Eusswehavel F. 4 Z. 3 L.as result; for^^x||x5= jff =l+^+T-f^.
The arguments are : — (thickness) 1 ZoE to 9 ZoU at intervals of | Zoll ;
(breadth) 1 Zoll to 18 Zoll at intervals of 1 Zoll; (length) 1 Fuss to 60
Fuss at intervals of 1 Fuss.

Another table (T. II.) is of the same kind, only intended for blocks &o. ;
so that the thickness is greater, and the result is only given in fractions of
a Kubik-Fuss.

T. III. contains volumes of cylinders for diameter (or circumference) of
section and length as arguments ; expressed as in T. I. and II. The money-
tables can have no mathematical value, as the Thaler = 30, 24, or 90
Groschen, &c.

T. X. is for the calculation of interest. The simple-interest tables (T. A)
are too meagre to be worth description. T. B and C may be described as
giving the compound interest and present value of £1 for any number of
years up to 100 at 3, 4, 5, and 6 per cent, per annum, visi.

i^ + mj ^""^ (^ + 1^)"'

to 6 decimal places.

Other tables of this kind that we met with have not been noticed ; the
title of one such is given under Jahn, 1837.

Hassler, 1830. [T. I.] Seven-figure logarithms of numbers from 10,000
to 100,000, with proportional parts. "The line is broken for the change in
the third figure, as in Callet.

[T. II.]. Log sines and tangents for every second of the first degree, to 7
places.

[T. III.] Log cosines and cotangents for every 30" of the first degree, to
7 places, with difierences.

[T. IV.] Log sines, cosines, tangents, and cotangents, from 1° to 3°, at
intervals of 10", with difi'erences, and from 3° to 45°, at intervals of 30", with
diff'erences for 10", to 7 places.

[T. v.] Natural sines^r every 30" of the quadrant, with difi'erences for
10", to 7 places.

Copies of this book were published with Latin, English, French, German,
and Spanish introductions and titlepages (the titles will be found in the list
at the end of the Eeport). The tables are the same in all ; and the special
titlepages for each table have the headings in the five languages. The
Eoyal Society's library contains the Latin copy perfect, and the introduc-
tions in the four modern languages bound together in another volume, pre-
sented to the Society by the author. At the end of the latter volume is
pasted-in a specimen page of the table, set up with the usual even figures ;

106 aEPORT~1873.

and the author has written on the hack, " This sheet proves that, with
the usual form of figures of the same size as those used in the tahles, they
•would not have heen distinctly legible." The figures actually used are very
thin, and have large heads and tails, resembling somewhat figures made in
writing ; and a comparison of the specimen and a page of the tables shows
very clearly the superiority of the latter in point of distinctness. The words
in minima forma are quite justified, as we do not think it would be possible
to make the tables occupy less room without serious loss of clearness. All
that is usually given in a page of seven-figure logarithms is here contained
in a space about 3 in. by 5 in. ; and yet, owing to the shape of the figures,
every thing is very distinct. The author says on the titlepage, " purgatce
ah erroribus prcBcedeniium tabularum;" but the last figure of log 62943
js printed 6 instead of 5. There is also another last-figiire error. See
f Monthly Notices of the Boy. Ast. See.,' March 1873.

A short review of this work by Gauss appeared in the ' Gottingische ge-
lehrte Anzeigen,' March 31, 1831 (reprinted ' Werke,' t. iii. p. 255).

Henrion, 1626. [T. I,] Logarithms to 20,001, to 10 places, with
interscript differences (characteristics not separated from the mantissas),
copied from Bbiggs, 1624.

[T. II.] Log sines and tangents for every minute, to 7 places (charac-
teristics nnseparated from the mantissae), taken from Guntbe, 1620. Hbn-
EioN had calculated some logarithms himself when he received Bbigss's work
(see Phil. Mag., Supp. No. Dec. 1872). The copy of Heneion we have
seen is in the Brit. Mus. The full titlepage is given in § 5.

Heutschen (Vlacq), 1757. [T. I,] Natural sines, tangents, and secants,
and log sines and tangents to every minute, to 7 places (arranged on what De
Morgan calls the Oellibrand model) (180 pp.), and [T. II.] logarithms of
numbers to 10,000, to 7 places, arranged in columns (100 pp.).

A former edition of 1748 is spoken of in the preface ; and it is stated that
the tables were compared with the editions of Vlacq, Leyden, 1651, the Hague,
1665, and Amsterdam, 1673. The type is very bold and clear, much easier
to read than in most modern tables.

This is one of the numerous series of small tables known by the name of
Vlacq, and is described here because it is not mentioned by De Morgan ;
small editions like the present are so difficult to meet with that it is desirable
to notice them whenever any are found.

Hobert and Zdeler, 1799. [T. I.] Natural and log sines, cosines, tan-
gents, and cotangents for the quadrant, divided centesimally; viz. these func-
tions are given for arguments from '00001 to '03000 of a right angle at in-
tervals of -00001 of a right angle, and from -0300 to -5000 of a right angle
at intervals of '0001, to 7 places, with differences. Expressed in grades (cen-
tesimal degrees) &c., the arguments proceed to 3" at intervals of 10", and
thence to 50' at intervals of V. The manner of calculation of the table
is fuUy explained in the introduction ; aiid this adds much to the value of the
work. Several of the fundamsnta were calculated to a great many places.
Two or three constants are given on p. 310,

B. Table of natural sines and tangents for the first hundred ten-thousandths
(viz. for -0001, -0002 &c.) of a right angle, to 10 places,

C. Tour tables, expressing (I.) 1°, 2°, 3° 89° (II.) 1', 2',. . . .59',

(III.) 1", 2",. . . .59", (IV.) 1'", 2'",, . . .59'", aU as decimals of 90°, to 14
places.

D. Three tables to express (I.) hundredths, (II.) thousandths, (III.) ten-
thousandths of 90°, in degrees, minutes, and seconds (sexagesimal).

ON MATIIEMATICAL TABLES. 107

_E. fovtv tables to express (I.) hours, (11.) minutes, (III.) seconds, (IV.)
thirds, as decimals of a day.

F. Small table to express decimals of a day, in tours, minutes, and
seconds.

Gr. Circular measure of "1, -2, . , , , -9, 1-0, of a right angle, to 44 places.

[T. III.] Logarithms of numbers to 1100, and from 999,980 to 1,000,021,
to 36 places.

The WQvV conclHdes"with two remarkable lists of errata found in the course
of the calculations, viz. 381 errors in the trigonometrical tables of Cahet, all
of which, with one exception, affect only the last figure by a unit, and 138
similar errors in Veoa's ' Thesaurus/ 1794. The errors in Callet have, we
presume, been corrected in the later Urates.

Hoiiel, 1858. T. I. Eive-figure logarithms of numbers to 10,800 with
the corresponding degrees, minutes an4 seconds, and proportional parts.
The constants 8 and T (see § 3, art. 13) are given at the top of the page ; ■
then follows a page of small tables for the conversion of degrees, minutes, &c.

T. II. Natural and log sines, tangents, and secants to every minute of the
quadrant, to 5 places, with proportional parts,

T. III. Gaussian logarithms. The addition and subtraction tables are sepa-
rated, as in Zbch (§ 4). In the first B is given for argument A, from A= -000
to 1"650 at intervals of '001, thence to 3-00 at intervals of -01, and thence
to 5-0 at intervals of "l. In the second B is given for argument C, from
C=-3000 to -4800 at intervals of -0001, thence to 1-500 at intervals of -001,
thence to 3*10 at intervals of '01, and to 5'0 at intervals of- •!, with pro-
portional parts : all to .5 places, These tables are followed by the first hun-
di'Od multiples of the modulus and its reciprocal, to 8 places,

T. lY. Tables to calculate logarithms to 8 places &e.

T, V. (one page). To calculate logarithms to gO places.

T. VI. A. page of four-figure logarithms to 600, and of three-figure anti-
logarithms.

T. VII. Least factors of composite numbers (not divisible by 2, 3, 5, or 11)
up to 10,841.

T, VIII. A page of constants. [We have since obtained a " nouvelle
edition, revue et augmentee," Paris, 1871, pp. 118 and introduction xlvi.]

Hijisse's Vega, 1840. T.I. Seven-figure logarithms to 1000, and from
10,000 to 108,000, with proportional parts ; the change in the line is denoted
by a small asterisk prefixed to the fourth figure of all the logarithms affected.
The portion from 100,000 to 108,000 is given to 8 places. On6_ page at
the end is devoted to a small table to convert common iAto hyperbolic seven-
figure logarithms, and vice versd,

T. II. Log sines, tangents, and arcs (all equal) to every tenth of a seoon^
to 1' ; log sines and tangents from 0° 0' to 1° 32' to every second ; log sines,
cosines, tangents and cotangents for every ten seconds from O'' to 6°, and
for every minute to 45° : all to 7 places. When the intervals are 10" or 1',
differences for a second are added : the logarithms are written at length,

The table is followed by a page giving the circular measure of 1°, 2°, 10°,

and thence by tens to 360°, of 1', 2',. . . .60', and of 1", 2" 60", to 11

places. . , , >,

T. III. Natural sines and tangents to every minute of the quadrant, to 7

places, with differences for 1". . , ■, -

T. IV, Ohord-table to radius 500, viz. lengths of semiohords of arcs

from 0° to 125° at intervals of 5', to 6 laces, for radius unity.

(^

e. Bin

108 REPORT— isrs.

This tabic is followed by 2 pages of tables for the conversion of centesimals
into sexagesimals «&c.

T. V. All prime divisors of numbers to 102,000 (multiples of 2, 3, and 5
excluded), and primes from 102,000 to 400,313.

T. VI. Hyperbolic logarithms of numbers to 1000, and of primes from
1000 to 10,000, to 8 places. This is followed by powers of 2, 3, and 5 to the
45tli, 36th, and 27th respectively.

T. VII. Powers of e and their logarithms, viz. ^ and log,„e% from a;=-01
to a;=10 at intervals of -01, to 7 figures and 7 places respectively.

T. VIII. Square and cube roots of numbers to 10,000, to 12 and 7 places

respectively. The table is followed by a page of coeflcients, such as h— ^

1 1.3

„ ■ „ , p— T — -, <fcc., to 10 places, and their logarithms to 7 places.

T. IX. Power-tables. A, the first 11 powers of numbers from -01 to 1-00
at intervals of -Ol, to 8 places. B, the first 9 powers of numbers from 1 to 100.
C, squares and cubes from 1 to 1000. D, the first hundred powers of 1-01, 1-02,
1-025, 1-0275, 1-03, 1-0325, 1-035, 1-0375, 1-04, 1-045, 1-05, 1-06, to 6 places.
E, the first hundred powers of the reciprocals of these numbers, to 7 places.
P, the sums of the powers in D : this table therefore gives a;+a;°+ . . . .»"

(-fe?)

for the values of x written down under D, and for w = 1, 2, 3, .

100. Gr stands in the same relation to E that P does to D. The tables from
D to G were calculated for their use in computing interest &c.

T. XII. An extended table of Gaussian logarithms. It gives B from A=>
•000 to A = 2-000 at intervals of -001, from A = 2-00 to A = 3-39 at intervals of
-01, and thencetoA=5-0 atintervalsof -1, toSplaoes. There are also given, be-
sides, other quantities for the same arguments, viz. C(=A-t-B), D (=B-i-C),
E (=A+C), and F (=B— A), all to 5 places, with differences and propor-
tional parts (of two kinds) for B and C.

„ „„^ , , , ,. , ,, . x(x—l) x(!e—V\....(!e—5) .
T. Xin. Interpolation table, viz. -^-5 — -, . . .— — =-^% ^ ~, from

a;=-01 to a;=l-00 at intervals of -01, to 7 places; then follows a page of
constants. There are, besides, mortality tables, very complete tables of mea-
sures and weights of different countries, &c. The table of 12-place square
roots was published' here for the first xime : it 'was calculated by Hensel in
1804. The seven-place cube roots, the chord-table, and the new columns of
the Gaussian table were calculated by Dr. Miohaelis, of Leipzig. The author
draws attention to the fact that the last figures in T. VIII. and XII. are given
correctly.

It is a matter of sufficient interest to note here that, though the work is called
an edition of Vboa, it contains one error from which the other tables known by
the name of Vega and published subsequently to his folio of 1794 were free.
In VLAca (1628), log 52943 was printed 7238085868 instead of 7238085468,
and the error was first pointed out and corrected by Vesa in his folio of 1794.
All the seven-figure tables, therefore, from 1628 to 1794 (and several of the
subsequent tables also), have 7238086 instead of 7238085 ; but Vega's small
editions (the ' Manuale ' and ' Tabulae ') have the logarithms correctly printed.
In HtTLssE's edition, however, the error is reproduced afresh, and the last figure
is printed 6. It follows therefore either that Hiilsse did not reprint Vega's
table, or that, if he did, he noticed the discrepancy, and decided in favour of
the erroneous value. The slight suspicion thus cast on these tables is unfor-

ON MATHEMATICAL TADLES. 109

tunate, as they form a most valuable collection, and are supplemental to
Cailet. We have seen advertised a second edition (1849) ; and Zeoh's tables
(see Zech, 1849, § 3, art. 19) are extracted from it. The last-figure error
noticed above is the only one of the hereditary Vlacq's errors that appears
in the table of the logarithms of numbers ; so that but for this curious
blunder the present work would have been, we believe, the first to
be free from errors of this class (see ' Monthly Notices of the Roy. Ast.
Soc' March, 1873). Some remarks by Gauss on T. XII. appear in t. iii.
pp. 255-257 of his ' Werke.'

Hutton, 1781 (products and powers of numbers). [T. I.] Products to
1000 X 100 (pp. 51).

[T. II.] Squares and cubes of numbers from 1 to 10,000 (pp. 54-78).

[T. III.] Squares of numbers from 10,000 to 25,400 (pp. 78-100).

[T. IV.] Table of the first ten powers of numbers from 1 to 100. Two
errors (viz. the last three figures of 81° should be 401, not 101, and the last
three of 98' should be 672, not 662) are poiated out by the reporter in the
Philosophical Transactions, 1870, p. 370.

The remaining three pages of the book are devoted to weights and mea-
sures &c. The table is closely printed; and some of the pages contain a great
many figures, as there are a hundred lines to the page. De Morgan states
that the table has not the reputation of correctness ; and the charge is no
doubt true, as, besides the two errors noted above (both of which we found
on the only page we have used), it is to be inferred from Baelow's intro-
duction to his tables that he found errors ; he did not, however, publish any
account of them.

Hutton, 1858. T. I. Seven-figure logarithms to 1000, and from 10,000
to 108,000, with proportional parts for all the diff'erences. The change in the
line is denoted by a bar placed over the fourth figure of all the logarithms
affected.

T. II. Logarithms to 1000, and thence for odd numbers to 1199, to 20
places.

T. III. Logarithms from 101,000 to 101,149, to 20 places, with first,
second, and third differences.

T. IV. Antilogarithms, viz. numbers to logarithms from -00000 _ to
•00149 at intervals of -00001, to 20 places, with first, second, and third
differences.

T. V. Hyperbolic logarithms from 1-01 to 10-00 at intervals of -01, and
for 10^ . . . lOS to seven places.

T. VI. Hyperbolic logarithms to 1200, to seven places.

T. VII. Logistic logarithms, viz. log 3600" -log a;, from co=l" to «=
5280" (=88') at intervals of 1", to four places, the arguments being ex-
pressed in minutes and seconds.

■ T. VIII. Log sines and tangents to every second of the first two degrees,
to seven places ; no differences.

T IX. Natural and log sines, tangents, secants, and versed sines for every
minute of the quadrant, with differences, to seven places, semiquadrantaUy
arranged. The natural functions occupy the left-hand pages, and the loga-
rithmic the right-hand. In both these last two tables the logarithms are all

written at full length. . . , f io go irao „f v o-

T XI. Circular arcs, viz. circular measure ot \ , 2, ,.. .. ioU , ot i , ^

60', of 1" 60", and of 1'" to 60'", to seven places.

' T. XII. Proportional parts to hundredths of 2-302 , the reciprocal of

the modulus.

110 REPORT— 1873.

Some constants are given in T. XX. ; the other tables consist of a traverse
table, formulae, &c.

The edition described above is one of those edited by Olintljus Gregory,
and is the last we have met with. The first edition was pubUshed in 1786,
the second in 1794, the third in 1801, the fifth in 1811, and the sixth, the
last published in Hutton's Hfetime (he died 1823), in 1822.

We have compared the first, second, and sixth editions, and that of 1858
described above. The first two are nearly identical, so that we need only
notice the differences between the tables of 1785; 1822, and 1858. In both
the two former of these editions T. I. only extends to 100,000 ; and while in
that of 1785 the change of figure in the line is not marked at all, in that of
1822 the fourth figure in the first logarithm affected only is marked. T. II. is
the same Ln the 1822 edition, but it ends at 1161 instead of 1199 in that of
1785. T. in. in 1785 ended at 101,139, and is extended to 101,149 in both
the other editions, as also did T. IV. originally end at '00139. In the edi-
tions of 1785 and 1822 occur two tables that were left out by Gregory in
1830 and in succeeding editions, viz. T. 5, giving logarithms of all numbers
to 100, and of primes from 100 to 1100, to 61 places, and T. 6, giving the
logarithms of the numbers from 999,980 to 1,000,020, to 61 places, with first,
second, third, and fourth differences. T. VI., of hyperbolic logarithms, ap-
pears in the edition of 1822, but not in that of 1785. T. VII. extended only
to 80' in 1785.

To aU the first six editions is prefixed Hutton's introduction, containing a
history of logarithms, the different ways in which they may be constructed,
&c. This very valuable essay was omitted by Gregory in the seventh (1830)
and subsequent editions (on account of its being rather out of place in a col-
lection of tables), and with some reason. In the 1786 edition it occupied
180 pp., 55 pp. of which are the " Description and Use of the Tables." This
portion Gregory retained j and in the 1858 edition it occupied 68 pp.

The whole work was reset in the later editions, published in Hutton's
lifetime, the chief additions, as we infer from the preface, having been made
in the fifth (1811) edition. On the last page of the 1822 edition are some
errata found in Cailet (1783, 1795, and 1801), and also in Taylor (1792) ;
the lists of errors in Gaiuiiner (London and Avignon) are also more complete
than in the earlier editions. Hutton's tables were the legitimate successors
of Sherwin's, and bring dovra to the present time one of the main lines of
descent from VnAca (see Shervtin, § 4).

Inman, 1871. [T. I.] Logistic logarithms, viz. log 3600'— log a; from a?
= 2 to a;=3600' (=60"') atintervals of 2', to 5 places. Arguments expressed
in minutes and seconds.

[T. II.] Proportional logarithms, viz. log 10800"— log x to every second
to 3° (same as T. 74 of Eaper, only to 5 places instead of 4), preceded by a
page giving the same for every tenth of a second to 1'.

[T. HI.] Log sines at intervals of 1" to 50', to 6 places.

[T. IV.] Log sines, tangents, and secants at intervals of 1' to 3'' (argu-
ments also given in arc, the intervals being 15"),' to 6 places: the table is
followed by a page of proportional parts for use with it.

[T. v.] J log haversines, viz. | log semi- versed sines = log sin -, from

a;=0° to 15° at intervals of 15", thence to 60° at intervals of 30", and
thence to 180° at intervals of 1', to 6 places (arguments also in time).
Note. — In several instances in this table ' is misprinted for ",
[T. VI.] Log Mversines. 8ame as previous table, except that 2 log sin

ON MATHEMATICAL TABLES. IH

- is the function tabulated; so that all the results are double those in [T. V.],

and that the intervals are 15" up to 135°, and then 1' to 180°.

[T. VII.] Six-figure logarithms to 1000, and from 1000 to 10,000 in de-
cades, with proportional parts.

[T. Vin.] Natural yersed sines to every second (of time) to 36"", to 6
places.

- JT. IX.] Natural versed sines to every minute (of arc) to 180°, to 6 places,
with complete proportional parts for every second up to 60". The other
tables are nautical.

The paging of the book runs at the top of the pages to 216, and thence at
the bottom to 275 ; it then recommences at the top at p. 217. This is no
doubt caused by [T. V., YI.] having been introduced in this edition only.

We have seen the original work, ' Nautical Tables designed for the use of
British Seamen, by James Inman, D.D. London, 1830' (400 pp. of tables),
but have not compared the two together : except for the " haversines," how-
ever, the tables seem to be nearly identical in the two editions.

Inman's ' Navigation and Nautical Astronomy ' (2nd edit.), Portsea, 1826,
contains no tables.

Irsengarth, 1810, These are merely land tables, and the units (Euth6,
Fuss, &c.) are so special that they do not appear to possess any mathema-
tical value.

Jahn, 1837. "Vol. I. Six-figure logarithms to 100,000 ; the change in
the line is denoted by a dagger (f) prefixed to the fourth figure of all loga-
rithms affected. There are no proportional parts on the page ; but they are
given in a separate table at the end.

Yol. II. Logarithmic sines and tangents for every second of the first
degree ; log sines and tangents for every third second of the quadrant (semi-
quadrantaUy arranged) : aU to 6 places. Proportional parts are given in the
extreme right and left columns of the double page for every twentieth of the
three-second interval.

The introductory matter is both in German and Latin.

We rather like the paper on which the second volume is printed ; though
not of a good quality, it is thick and stiif, and of a brownish colour, so that
the book could be, we think, used for a long time at once without injury to
the eye : the first volume (in the copy before us), however, is printed on
paper of the soft, flaccid kind common in German books.

The author was led to publish his tables by observing that nearly all those
in use were either five- or seven-figure tables.

We have seen, by the same author, ' Tafeln zur Berechnung fUr Kubik-
Inhalt &c.,' 2nd edit., Leipzig, 1847 ; but the tables are commercial (argu-
ments expressed in ZoUe, EUen, &c.), and do not need notice here.

Kerigan, 1821. T. YIII. Log sines for every second to 2°, and thence,
at intervals of 5", to 90°, to six places ; in this latter part of the table pro-
portional parts for seconds are added, so that the table -practically gives log
sines to every second; arranged quadrantally. The logarithms are all printed
at length.

T. IX. Natural sines from 0° to 90° at intervals of 10", to six places;
no differences ; the sines written at length.

T. X. Six-figure logarithms from 1000 to 10,000, with proportional parts ;
arranged as is usual in seven-figure tables ; the change in the line is
marked by the ciphers after the change in the third place being filled in,
so as to render them black circles.

112 REPORT— 1873.

T. XI. Logarithmic Rising, viz. log versed sines from 0* to 8'' at inter-
vals of 5', with, proportional parts to seconds, to 5 places : the logarithms are
written at length.

T. XII. Proportional logarithms for every second to 3°, to four places ;
same as T. 74 of Eapee.

T. XIIL Small table to convert arc into time : the other tables are
nautical.

Kbhler, 1832. [T. I.] Five-figure logarithms to 10,000, arranged con-
secutively in columns, with differences and characteristics ; the degrees, min-
utes, &c. for every thirtieth number are added.

[T. II.] Log sines and tangents for every minute of the quadrant, to five
places, with differences.

[T. III.] Gauss's table (§ 3, art. 19) ; viz. B and C are given for argument A
from -000 to 2-000 at intervals of -001, thence to 3-40 at intervals of '01,
and to 5 at intervals of -1, to five places, with differences.

There are besides a few constants; the introduction is in French and
German.

Kohler, 1848. [T. I.] Seven-figure logarithms to 1000, and from 10,000
to 108,000 (this last 8000 being to 8 places), with differences and proportional
parts ; the change in the line is denoted by a bar placed over the fourth figure of
all the logarithms affected. The constants S and T (§ 3, art. 13) and the
variation are given at the top of the page, as also is the number of degrees,
minutes, &c. corresponding to every tenth number. At the end are the first
hundred multiples of the modulus and its reciprocal to 8 places, and a small
table to convert arc into time.

[T. II.] Gaussian logarithms : B and C are given to 5 places (with differences)
for A = -000 to 2-000 at intervals of -001, thence to 3-40 at intervals of -01,
and to 5-0 at intervals of -1 (same as Gauss's table 1812, § 3, art 19).

[T. III.] Briggian logarithms of primes from 2 to ISll, to 11 places, fol-
lowed by 2 pages of constants, some weights and measures, &c.

[T. IV.] Log sines, tangents, and arcs (all equal) for every second to 1' ;
and log sines, cosines, tangents, and cotangents for intervals of 10" to 10°,
and thence for intervals of 1' to 45°, to 7 places, with differences for one
second.

[T. v.] Circular measure of 1°, 2° 100°, 110° 300°, 330°, 360°,

of 1', 2' 60', and of 1", 2" 60", to 11 places. Then foUow some for-
mulae, and we come to the second part of the work, ' Mathematische Tafeln,
die oft gebraucht werden,' containing : —

T. I. Hyperbolic logarithms (to 8 places) of numbers from 1 to 1000,
and of primes from 1000 to 10,000.

T. II. The first 45, 86, and 27 powers of 2, 3, and 5 respectively.

T. III. e^ from a,-=-01 to 10-00 at intervals of -01 to 7 figures.

T. IV. The first ten powers of numbers from 1 to 100.

T. V. Squares of numbers from 1 to 1000.

T. VI. Cubes of numbers from 1 to 1000.

T. VII. Square and cube roots (to 7 places) of all numbers from 1
to 1000.

T. VIII. Factor tables, giving all divisors of all numbers not prime or
divisible by 2, 3, or 5, from unity to 21,524.

T. IX. To express minutes and seconds as decimals of a degree &c.

T. X. Binomial-theorem coefficients, viz. x, <^~'^\ &e. . . .?_i^- G'g-4)

1.2 ' 1.2.3.4'

from x= -01 to 1-00 at intervals of -01, to 6 places.

ON MATHEMATICAL TABLES. 113

1.3.5 1

T. XI. Decimal values of certaiin coefficients, sucli as

.4.6.7 2.4.5'

1.3

„ ■ ' ^ ■ , &c., -with their logarithms. There are 40 in aU ; and the table

occupies one page.

A reward of a louis d'or was offered for every error found in the first
edition ; all the errors so found are corrected in the second, here described.

Lalande, 1805. [T. I.] Five-figure logarithms of numbers from 1 to
10,000, arranged consecutively in columns, with differences.

[T. II.] Log sines and tangents for every minute of the quadi-ant, to 5
places. An explanation of 34 pp. is prefixed.

Iialaude, 1829. [T. I.] S.even-figure logarithms to 10,000, arranged iu
columns with characteristics and differences ; the number of degrees, minutes,
&c. for the first number in each column (viz. for every thirtieth .number) is
given at the top.

[T. II.] Log sines and tangents for every minute of the quadrant, to 7
places, with differences.

Lambert, 1798. T. I. Divisors of all numbers up to 102,000 not divi-
sible by 2, 3, or 5. If the number is the product of only two prime factors,
then the least only is given ; but if of more than two, the others are given,
except the largest. The table therefore gives all the simple factors except
the greatest. The letters/, g, h, &c. are used for 11, 13, 17, &c. (as explained
on p. xviii of the introduction), not only because they occupy lesrf room, but
also because they can be placed in contact without risk of mistake ; the
least factor, however, is always written at length.

T. II. Abacus numerorum primorum, viz. first. 10 multiples of all the
primes up to 313.

T. III. Seven products, each of seven consecutive primes, from 7 to 173.

T. iV. List of the three-figure endings that squares of odd numbers
admit of.

T. VI. Primes from 1 to 101,977.

T. VII.-IX. Powers of 2 to 2'°, of 3 to 3'°, of 5 to 5".

T. XL «-^ (to 7 places) for x=-l, % . . . -9, 1, 2, . . . 10.

T. XIII. & XV. Hyperbolic logarithms (to 7 places) of numbers from
1 to 100, and from 1-01 to 10-00 at intervals of -01, respectively.

T. XIV. & XVI. contain log^ 10, 10' . . . 10", to 7 places, and log^ 2,

3 ... 10, and 1 vTT, to 25 places.

' log« 10'

T. XVII. Tables of numbers of the form 2". 3'\ 5". 7' arranged in order
up to 11,200.

T. XXIII. Circular measure of 1°, 2°. . .100°, 120°, 150°, 180°. . .360°,
of 1', 2' . . .10', 20' . . .60', and of 1", 2" . . .10", 20" . . .60", to 2'r places.

T. XXIV. 0=lOOOO"m; 0, (^\ . .V expressed in terms of in (in circular
Ineasure), to 16 places, and sin ^, cos ^ expressed in terms of m with decimal

coefficients, to 18 places. Also tt, log x, -, ^J^^, &c. to a good many places.

T. XXV. Natural sines to every degree and their first 9 multiples, to 5
places.

T. XXVI. Sines, tangents, and secants, and log sines and tangents to
every degree, to 7 places.

T. XXIX. Table for facilitating the solution of cubic equations, viz.
«= +(x—!^) from a;='001 to 1-155 at intervals of -001, to 7 places.

1873. I

114

KEPORT — 1873.

T. XXXII. Funetiones liyperholicm eircularibits atialogm, Q g being a
rectangular hyperbola, centre C, P C Q is the so-caUed angulus tramcendens
= (j, say, y C G thp angulus communis =4» say; j) g is the hyperbolie sine,
Cjp the hyperbolic cosine, and C g Q the sector; so that if the hyperbola be
a;''—y'=l, a;=6ec ^ and y=tan ^,

The argument is 0, and proceeds from 0° to 90° at intervals of 1° ; and
the table gives the sector, y, ce, log y, log x, tan \p, log tan ip and i//, all ex-
cept the last to 7 places, and the last to one decimal of a.second.

T. XXXV. & XXXVI. Squares and cubes of numbers from 1 to 1000.

(first 12 series), viz. «,

T. xxxvn,

a:(.r+l)(ig+2)

Figurate numbers
«;"(;» +1).. .(a; +11

1.2

from jB=l to 30,

1.2.3 ' 1.2.3... 12

T. XI, First 11 powers of -01, -02, -03. . .1-00, to 8 places.

T. XLIV, Coefficients ofthe first 16 terms in (l+x)^ and (l + a;)"*, their
accurate values being given as decimals.

Besides the above, T. XIX. gives sin 3°, 6° . . .89° in radicals, and T. XLII.
the fii-st 6 or 9 convergents to ^2, V3, s/5.,,i<J12 as vulgar fractions.
The other tables contain formidae &o.

The work is edited by Felkel, who has prefixed a Prmfatio Interpretts of
xi pp., giving a description of his (Felkel's) tables of divisors &o. ; and there
is also added at the end an account of his proposed scheme of tables in rela-
tion to the theory of numbers. About Felkel, see Fblkel, 1776, § 3, art, 8.

The titlepage states that this is a translation from a German edition. The
original was entitled " Zusatze zu den logarithmisehen und trigonometrisohen
TabeUen," and was published in 1770 ; or, at all events, De Morgan's descrip-
tion of the contents of this latter work, which we have not seen, agrees,
as far as it goes, almost entirely with the ' Supplementa ' &o., which De Morgan
had heard of, but not seen. The introduction to the latter shows signs of
having been ataplified by Felkel.

Lax, 1821. T, XIV, Proportional logarithms, viz, log 10800" — log «>
from i»=0" to a;=10800" (=3°) at intervals of 1" (the arguments being
expressed in degrees, minutes, and seconds), to five places. On the first page,
however, which extends to 10', only two, three, or four places are given cor-
rectly, the number being filled up to five by adding ciphers j facing 0° 0' 0"
there is given 4*88 . . instead of — oo .

T. XVII. Natural versed, suversed, coversed, and sucoversed sines, viz.
1— cos x and 1 + cos « for every minute of the quadrant, to six places, with
proportional parts for 1", 2", . . 60", so that the tabular results can bo taken
out very easfly to seconds. It may be observed that of the double columns

ON MATHEMATICAL TABLES. 115

headed ' and " the iirst refers to the argument and the second to the propor-
tional parts. This table occupies pp. 57-80 of the book.

T. XYIII. Six-flgure logarithms to 15,500, ■vrith proportional parts at
the foot of the page to twentieths for the portion beyond 1000. The table is
so arranged that all the logaiithms are given at full length, though this is
not the case with the numbers ; for example, to find the logarithm of 15184
we seek 15150 at the head of the column, and line 34 in the column : this
defect might have been partially remedied by the introduction of another
column at the right-hand side of the page containing the numbers, 50,
51 . . . 99. The other tables, 22 in number, are nautical.

Lrynn, 1827. T. Z. (pp. 244-283). A sexagesimal proportional table,
- exhibiting at sight, in minutes, seconds, and tenths of a second, the fourth
term in any proportion in which the first term is 60 minutes, the second term
any number of minutes under 60 minutes, and the third term any number of
minutes and seconds under 10 minutes. If the second term is not an exact
number of minutes the table can still be used, though two operations are

required. The table may be described as giving -^, in minutes, seconds, &c.,

X (running down the column) being 1', 2' . . . 60', and y (running along the
top lines) extending to 10' at intervals of 1".

T. E. (pp. 288, 289). Proportional logarithms for every minute to 24'',
viz. log 1440"" -log a-, from a;=l"" to «=1860'" (=3r')at intervals of unity,
the arguments being expressed in hours (or degrees) and minutes, to four
places ; the other tables are nautical.

Mackay, 1810 (vol. ii.). T. XLI. Natural versed sines for every ten
seconds to 180°, to six places.

T. XLY. Six-figure logarithms of numbers to 100, and from 1000 to
10,000, with difi^erences ; the logarithms written at length.

T. XL VI. Log sines to every ten seconds of the quadrant, to six places.

T. XL VII. Log tangents to every ten seconds of the quadrant, to six places,

T. XLVIII.-L. To find tlie latitude hy double altitudes of the sun or stars
and the elapsed lime. The first and second of these tables give log cosec on
and log (2 sin x) from x=(i^ to ^■=3'' 59" 50^ at intervals of 10* ; and the
third gives^^log versed sines to 7'' 59"' 50' at intervals of 10^ aUto five places,
the logarithms being written at length. These tables were copied, according
to the author (see note, vol. ii. p. 31), from the second edition (1801) of this
work witlioiit acknowledgment into Noeie's ' Epitome of Navigation.'

T. LI. Proportional logarithms to every second to 3°, to four places ; same
as T. 74 of Eaper ; the other tables are nautical.

The table of natural versed sines was calculated for this work, and ap-
peared in the first edition (1793) ; it has since, the author states, been fre-
quently copied (see note, vol. ii. p. 13).

Maseres, 1795. This is a collection of reprints of tracts, and, among
others, of " An Appendix to the English Translation of Ehonius's German
Treatise of Algebra, made by Mr. Thomas Brancker, M.A., ... At London, in

the year" 1668 " And on pp. 367-416 is given "Thomas Brancker's Table

of Incomposit or prime Numbers, less than 100,000," viz. least factors of all
numbers up to 100,000 not divisible by 2 or 5. On p. 366 is a rather long list
of errors in the table (we suppose Maseres reprinted verbatim from his copy,
as somfe of the errata are corrected and some are not), and also some errors
in Guldinus, Schooten, and Ehoiiius. The table is preceded (pp. 364, 365)
by ' A Tarriffa, or Table, of all Incomposit or prime numbers less than
VlOOjOOO, multiplied by 2, 3, 4, 6, 6, 7, 8, 9,"

i2

116 iiEPORT— 1873.

Ou pp. 591, 592, T. XIX. of Dodson's ' Calculator,' 1747 (viz. square and
cube roots of numbers less than 180, to 6 places), is reprinted ; and on pp.
595-604 are reciprocals (to 9 places) and square rools (to 10 places) of
numbers from 1 to 1000, reprinted (as Maseres states in the preface) from
vol. iv. of Mutton's 'Miscellanea Mathematica' (1775, 4 vols. 12mo).

Maskelyne (Eequisite Tables), 1802. T. XV. Proportional logarithms
for every second to 3°, to 4 places ; same as T. 74 of Kapeb.

T. XVI. For eomputhui the latitude of a ship at sea, &c. The arguments run
from 0" to e"" at intervals of 10'; and there are three columns of tabular results
headed Log | Elap. time, Log Mid. time. Log rising, which give respectively
log cosec X, log (2 sin a,-), and log vers sin x, to 5 places ; the log rising is
also continued for arguments from 6'' to 9'' at the same intervals. This table,
modified in form &c., is reproduced in Mackat, Domke, (fee. (see § 3, art. 15,
p. 68, and Bowditcf, 1802), and is sometimes called by Maskelyne's name.
T. XVII. Natural sines to every minute of the quadrant, to 5 places.
T. XVIII. Five-figure logarithms of numbers to 10,000.
T. XIX. Log sines, secants, and tangents to every minute of the qua-
drant, to 5 places; the sines are given to 6 places, the last being separated
from the rest by a point ; the other tables are nautical.

Maskelyne's name does not appear on the titlepage to these tables ; but
the preface is signed by him.

Appendix to the Thied Edition. T. I. Natural sines to every minute
of the quadrant, with proportional parts for seconds.

T. II. Natural versed sines for every minute to 120°, with pi'oportional
parts for seconds.

T. III. Logarithms of numbers to 1000, arranged consecutively, and
printed in groups of five ; and thence to 100,000 grouped in decades, with
proportional parts for each decade by its side. All the tables in the Appen-
dix arc to six places. Copies of the Appendix were circulated separately.

Minsinger, 1845. [T, I.] Seven-figure logarithms to 100 and from
1000 to 10,000, with proportional parts at the foot of the page ; the sixth
place is separated by a comma from the seventh, for convenience if the table
is to be used to six places. The change in the line is denoted by an asterisk
attached to all the logarithms affected.

[T. II.] Squares, cubes, and square and cube roots (to 6 places) of all
numbers from 1 to 100, and squares and cubes only of numbers from 100 to
1000. Then follow a few constants and [T. IV.] primes to 1000.

Moore, Sir Jonas, 1681. [T. I.] Seven-figure logarithms to 10,000
(arranged as is now usual), with differences : the proijortional parts [T. II.]
»re given by themselves at the end, and occupy 22 pp. This may be regarded
as a separate table, containing proportional parts (to tenths) of numbers
from 44 to 4320— the interval being 2 to 900, 3 to 999, 4 to 1415, 5 to 2000,
and 10 to 4320.

[T. III.] Natural and log sines, tangents, and secants to every minute of
the quadrant, to 7 places (semiquadrantally arranged), without differences.
It may be remarked that many of the N's at the top of the columns are
imperfectly printed, and appear like V's ; thus N, tangent is often printed
V. tangent.

[T. IV.] (pp. 262-351). Natural and log versed sines from 0° to 90° to

every minute, to 7 places. De Morgan says that this is the first appearance of

this table in England. The other tables relate to navigation, geography, &c.

[Moore, Sir Jonas, 1681] (Versed sines). Natural and log versed sines

to every minute of the quadrant, to 7 places, semiquadrantally arranged.

ON MATHEMATICAL TABLES. 117

. The copy of this tract before us (which is bound up in a volume with
several others, and belongs to the Cambridge University Library) is clearly
either a separate reprint or merely a table torn out from some larger
work. The paging runs from 262 to 351 : at the beginning there is a plate,
the size, of. the page, of a person observing -with a sextant, and the words
" between page 248 and 249 " below in the left hand-corner, and at the end
a diagram with a movable circle and pointer, headed " The fore part of the
. Nocturnall or side held next the face in time of observation," and " between
page 254 and 255 " below. On examination we find the table is [T. IV.] of
Sir Jonas Moobe's ' Systeme of the Mathematicks,' 1681, just described.
The engravings do not, however, appear to be taken from either volume
of this work. It is very likely that this table was merely torn out
from the work, and was never published separately ; stiU as, according to
De Morgan, this is the first appearance of such a table in England, it is not
improbable that copies may have been in request, and therefore issued
separately.

J. H. Moorej 1814. T. III. Log sines, tangents, and secants to every
quarter-point, to 5 places.

T. IV. Eive-figure logarithms of numbers to 10,000.

T. V. Log sines, tangents, and secants for every minute of the quadrant, to
5 places,

T. XXIII. Log ^ elapsed time, mid. time, and rising (for explanation of
these terms see T. XVI. of Maskeltue, § 4) for every 10' to 6'', except
the last, which is to 9'', to 5 places. The tables are separated as in Mackat.

T. XXIV. Natural sines for every minute of the quadrant, to 5 places.

T. XXV. Proportional logarithms for every second to 3°, to 4 places ; same
as T. 74 of Eapbb.

We have seen the 18th edition (1810), which, is identical with that above
described, an edition of 1793, and the 9th edition (1791) (the last two not
edited by Dessiou). All contain the tables described in this account (though
the order is different)," except that the tables in T. XXIII. are not separated;
the log rising is only given to 6**, and the intervals also 30', in the two
earlier editions.

Three out of the four editions contain different portraits of the author.

MtUler, 1844. [T. I.] Pive-figure logarithms of numbers from 1000 to
1500, and four-figure logarithms from 100 to 1000.

[T. II.] Table of Gaussian logarithms in a somewhat modified form,
viz. S and U to 4 places, from A=-0000 to -0300 at intervals of -0001,
thence to -230 at intervals of -001, and from -20 to 2-00 at intervals of -01,
and thence to 4-0 at intervals of -1, with differences ; where

A = log x, S = log Tl -f- -Y and U = log .

a;

[T. III.] Squares of numbers from to 1 at intervals of -0001, to 4 places,
and quarter squares of numbers from to 2 at the same intervals, also to 4
places (intended for use in the method of least squares).

[T. IV.] Four-place log sines and tangents for every second to 10', thence at
intervals of 10" to 1°, thence at intervals of 1' to 4°, and to 90° at intervals
of 10'.

There are given also ; — the circular measure (to 12 places) of 1°, 2° . . .
10°, 1' . . . 10' and 1" . . . 10" ; 12 constants involving tt ; natural sines and
tangents to every half degree ,' and a few three-figure logarithms.

118 BEPORT— 1873.

John Newton, 1658. [T. I.] Logarithms to 1000, to 8 places, and
logarithms from 10,000 to 100,000, also to 8 places. A column is added to
each page containing the logarithms of the differences, to 5 places.

[T. II.] Log sines and tangents (semiquadrantally arranged) for every
centesimal minute (viz. nine-thousandth part of a right angle), to 8 places,
with differences.

[T. HI.] Log sines and tangents for the first three degrees of the quadrant,
to 5 places, the interval being the one thousandth part of a degree. Loga-
rithms of the differences to 8 places arc added.

■ The trigonometrical tables are thus of the kind introduced by Beiggs, and
are partly centesimal (see § 3, art. 15, p. 64). This is th« only extensive
eight-figure table that has been published ; and it is also remarkable on
account of the logarithms of the differences, instead of the differences, being
given. It seems worth consideration whether, in the event of a republication
of ViAca, 1628, it would not be advantageous to replace the differences by
their logarithms. It is usually most convenient, if many logarithms are to
be taken out at one time, to interpolate for the last five figures in a ten-
figure table by means of an ordinary seven-figure table ; but in other cases
recourse is generally had to simple division, and the natural differences are
best. The table would occupy too much space if both the differences and
their logarithms were added ; and there is not much chance of two publi-
cations ever being made, one with natural, and the other with logarithmic,
differences. If the choice had to be made, the decision would probably be in
favour of the simple differences as they are, though a good deal might be
urged on the other side.

A few errata are given at the end of the address to the reader, and a great
many more on the last page ; the tables, however, reproduce nearly all
VLACa's errors, which affect the first 8 places (see ' Monthly Notices of the
Eoy. Ast. Soc' March 1873). This was the first table in which the arrange-
ment, now universal in seven-figure tables (viz. with the fifth figures run-
ning horizontally along the top line of the page), was Used. The change of
the third figure in the line is not noted.

The title of this work being the ' Trigonometria Britannica ' (printed
' Britaniea ' on the titlepage), it is often confounded with Bkiggs's work of
this name, Gouda, 1633 (§ 3, art. 16), from which it is derived. Also, as
Gellibrand's name appears on the titlepage it is sometimes attributed to
him in catalogues.

In the Cambridge University Library is a copy of this book, in which the
titlepage and introduction are absent, the first page being the titlepage to
the tables, so that the work is anonymous. Whether some copies of the tables
alone were published, or whether the copy in question is imperfect, we do not
know.

Norie, 1836. T. XXIII. Log sines, tangents, and secants to every quar-
ter-point, to 7 places.

T. XXIV. Six-figure logarithms of numbers to 10,000, with differences.

T. XXV. Log sines and tangents to every ten seconds to 2°, and log sines,
tangents, and secants for every minute of the quadrant, to 6 places, with
differences.

T. XXVI. Natural sines for eveiy minute of the quadrant, to 6 places.

T. XXVII.-XXIX. To find the latitude by double altitudes and the
elapsed time. Log ^ elap. time, middle time, and rising (for explanation of
these terms sec T. XVT. of Maskklyne, § 4) are given at intervals of 5',
the two former to Q\ and the last to W^, to 5 places, with proportional

ON MATHEMATICAL TABLES. 119

parts. The three tables are separated, as is now usual (see Maceay, S 4,
T. XLVIII.). r > » K > s >

T. XXXI. Logarithms for finding the apparent time or horary angle,

VIZ. log _ f^ = 2 log siii^ j from oo ■= 0" to x = 9" at intervals of

5% to 5 places, with proportional parts for seconds.

T. XXXIV, Proportional logarithms for every seeond to 3°; same as
T. 74ofEAPEE.

T. XXXVI. Natural versed sines to every minute of the quadrant, with
proportional parts for every seeond of the minute-interval, to 6 places.

The other tables are nautieaL These tables also appear in Noeie's ' Epi-
tome of Navigation.'

"Narie (Epitome), 1844. The tables are the same as in Norm's Nautical
Tables just described ; they are added after the explanatory portion, which
occupies 328 pp.

On the different editions, see Noeie's Epitome in § 5.

BTorwood, 1631. Seven-figure logarithms to 10,000, and log sines and
tangents to every minute, to 7 places, semiquadrantaUy arranged: of the
latter we have seen separate copies under the title, " A triangular canon
logarithmicall " (the title it has also in the work). The editions we have
seen are :— first, 1631 ; second, 1641 ; third, 1656 ; seventh, 1678.

This was one of the first small tables in which the trigonometrical eanon
was derived from ViACft, 1628, and not GuntEb, 1620.

Oppolzer, 1866. Four-figure logarithms, with .proportional parts to
1000. A page of Gaussian logarithms, after Filipowski, and a page of pro-
portional parts. Log sines, cosines, tangents, cotangents to 10° at intervals
of 1', with diiferences, and from 10° to 45° at intervals of 10', with differ-
ences and proportional parts, all to 4 places.

Oughtred, 1657. [T. I.] Sines, tangents, and secants (to 7 places) and
log sines and tangents (to 6 places) for every centesimal minute ( = -^^-^ of a
right angle) of the quadrant. Sines, tangents, and secants on the left-hand
page of the opening, and cosines, cotangents, and cosecants, &c. (though not
so called or denoted) on the right-hand page.

[T. II.] Seven-figure logarithms of numbers from 1 to 10,000, followed
by a ' Tabula differentiarum ' for the sines and tangents.

In an appendix at the end of the book it is explained that the logarithmic
sines and tangents were intended by the author to consist of seven figures
after the index, but that " the seventh figaire was unhappily left out." This
is also referred to in the dedication.

Ozanam, 1685. Natural sines, tangents, and secants, and log sines and
tangents, and logarithms of numbers to 10,000, all to 7 places. There are
120 pp. of trigonometry &c. De Morgan points out that the tables are really
Vlacq's, though his name is not mentioned, and takes occasion very truly to
remark how many authors have considered that the merit of their books con-
sisted in the trigonometry, and that the tables (which usually form by far the
greater part of the work) were accessories of which no notice need be taken.

Parkhurst, 1871. This little book contains forty-two table_s, with the
last two of which this Eeport is not concerned. In describing briefly their
contents, it will be convenient to mention first the tables which contain
results most common in other works, such as logarithms &c., viz.: — ■
T. II., III., and IX. Logarithms from 1 to 109, to 102 places.
T. V. Multiples of the modulus -43429 . . . from 10 to 96, to 35 places.
T. XII. Logarithms of numbers from 1000 to 2199 at intervals of unity.

120 REPOKT — 1873.

from 2200 to 2998 at intervals of 2, from 3000 to 4995 nt intervals of 5;
all to 10 places (from ViAca).

T. XIII. Logarithms of numbers from 200 to 1199, to 20 places (from
Cailee).

T. XIV. (continuation of T. XIII.). Logaiithms of numbers from 1200
to 1399 at intervals of unity, from 1400 to 2998 at intervals of 2, from
3005 to 4995 at intervals of 10 ; all to 20 places.

T. XVIII. Logarithms of primes from 113 to 1129, to 61 places (from
Callet).

T. XX., XXI., XXII. A table of least divisors of numbers to 10,190,
and, for certain divisors, to 100,000. Multiples of 2, 3, 5, 7, aud 11 are
excluded ; it is very inconveniently arranged, and is moreover imperfect.

T. XXIII. Primes to 12,239.

T. XXV. Reciprocals from 300 to 3299, to 7 places, arranged like an ordi-
nary table of seven-figure logarithms.

T. XXVI. Products of the numbers from 200 to 399 by the digits 1, 2 ... 9,
and squares from 200^ to 399°.

T. XXVII., XXVIII. A few logarithms and antilogarithms, to 3 places,
and a similar small table to 4 places.

T. XXX., XXXI. Natural and log sines and tangents &c., to 4 places.

T. XXXII. Binomial-theorem coefficients (the first six for indices from
unity to 40), and squai'es from I'' to 200".

T. XXXIII., XXXIV. Multiplication table from 16 x 13 to 99 x 98,
and multiplication table of squares from 16^ x 13 to 99^ x 98.

T. XXXV., XXXVII., XXXVIII. Antilogarithms, logarithms to 8 places,
and log sines.

The other tables are : —

T. IV. Logarithms of factors, 102 decimals. T. VI. Secondary multi-
ples. T. VII. Factors to 3 decimals. T. VIII. Logarithms of factors, 31
decimals. T. X. Factors to 61 decimals. T. XI. Log F, for logarithms to
10 decimals. T. XV., XVI., XVII. Logarithms to 20 decimals of factors.
T. XIX. Constants derived from the modulus. T. XXIV. Log p, for addition
and subtraction. T. XXIX. Subtraction logarithms. T. XXXVI. Factors.
T. XXXIX., XL. Interpolations, Bessel's coefficients.

Most of these tables are tabulated for their use in the calculation of
logarithms by well-known methods. The arrangement of the work is most
confused ; and it would be very difficult to understand from the author's
description the objects of his tables. The paging of the book runs from 1
to 176 ; and this portion includes all the tables. Then Part 2 commences,
and the pages are numbered afresh from 1 to 38. In Part 3 the pages pro-
ceed from 1 to 27. Parts 2 and 3 arc occupied with a description of tho
tables ; and the reader who wishes to understand the meaning of the nota-
tion (which is often needlessly complex and confusing, to save the space of
a few figures), &c., is recommended to begin at Part 3, p. 5. It would take too
much room, even if it were worth while, to explain the tables in detail ; but it
may be stated that the tables (for the construction of logarithms of factors) give

the values of log n -i-^ Und log 1 1 — ''* 1 for different values of m and n

\ lO"/ ^ V 10"/

to a great many places, as required in Weddle's and similar methods.

It will save the reader some trouble to mention that by " o w in the
book is meant log (1-1- yt:„|, and by -7i °m, — log ll—^j. Generally

ON MATHEMATICAL TABLES. 121

the m is left out, where it is thought the context prevents risk of mistake ;
and instead of — n o m there is sometimes written nim, and the heading
" cologarithm." The jast page of the book, headed (wrongly) Table XXXIII.,
contains a very imperfect list of the abbreviations used.

It is to be inferred from the Preface &c., that the book was set up and
electrotyped by the author himself, who states that " it is probable that there
is not now a single eri-or in the whole table." The reward of a copy of the book
is also offered to the first finder of any important error under certain condi-
tions. Parts of the book, in the copy before us, are very badly printed, so
badly in fact that one or two pages are wholly illegible ; and the tables are so
crowded that we should think no one would use them who could procure any
others that could be made to do as weU. In fact the author's object seems to
have been to crowd the greatest possible amount of tabular matter into the
smallest space, without any regard to clearness. It is stated in the work that
in the course of the printing, incomplete copies (some containing proofs almost
illegible) were distributed to the author's friends ; and an advertisement on the
cover states that copies containing proofs rejected in the printing may be had
at different prices according to their completeness and the order of the tables.
The book is printed phonetically ; and this adds to the awkwardness of the
most confused, badly printed, and ill-explained series of tables we have met
with in the preparation of this Eeport. By issuing his tables in the form
and manner he has adopted, the author has not done justice to himself, as
several are the results of original calculation and are not to be met with
elsewhere.

Pasquich, 1817. T. I. Five-figure logarithms to 10,000 (arranged
consecutively in columns), without differences.

T. II. Log sines, cosines, tangents, and cotangents, from 0' to 56' at in-
tervals of 10", thence to 1° at intervals of 20", and thence to 45° at intervals
of 1', with differences for 1". Also squares of natural sines, cosines, tangents,
and cotangents from 1° to 45° at intervals of 1', all to 5 places. De Morgan
says, " This trigonometrical canon in squares is, we suppose, almost unique."
T. III. Gaussian logarithms. B and C (same notation as in Gitrss), to 5
places, with differences, for argument A, from A=-000 to A=2-000 at
intervals of -001, from A = 2-00 to A = 3-40 at intervals of '01, and from
A = 3-4 to A = 5 at intervals of -1. This table is the same as that originally
given by Gattss, 1812 (§ 3, art. 19).

A few constants <fec. are added in an Appendix.

A lengthy review of this work by Gauss appeared in the ' Gottingische
gelehrte Anzeigen' for Oct. 4, 1817. It is reprinted on pp. 246-250 of
t. iii. of his ' Werke.'

Pearson, 1824. Vol. I. contains 296 large quarto pages of tables ; but
only three pages come within the range of this Eeport, viz.: — [T. I.], p. 109,
a one-page table to convert space into time, and vice versa. [T. II.], p. 261,

which expresses 1°, 2°, 3° 360°, and 1', 2' 60' as decimals of the cir^

cumference of the circle to 4 and 5 places respectively ; and [T. III.], p. 262,

which gives the circular measure of 1°, 2° 180°, of 1', 2' 60' and of

1", 2" 60", to 8 places.

The other tables are nautical, astronomical &c.

Peters, 1871. FT. I.] pp. 16, 17. Hundredths, thousandths, ten-thou-
sandths, hundred-thousandths and millionths of a day expressed in minutes
and seconds.

[T. II.] pp. 18, 19. For the conversion of arc into time, and vice versd.

122 REPORT— 1873.

[T. III.] pp. 20, 21. Lengths of circular arcs, viz. 1°, 2°, 3°.... 90°,

thence to 115° at intervals of 5°, and to 360° at intervals of 10°, 1', 2' 60',

and 1", 2" . . . . 60", expressed in circular measure, to 7 places.

[T. IV.]. Interpolation tables. Table I. (p. 103) gives fH^^JlJ),

w(^ - 1)(^ - I) a„d (^ + IM^ - 1X^-2) f,,^ ^^.00 to ^.=1-00 at

48

intervals of -01 — the first function to 5 places (with differences), and the
second and third to 4 places (without differences). It will be noticed that on
writing 1 — a; for x, the first and third functions are unaltered, while only
a change of sign is produced in the second. It is thus sufficient to tabulate
them only from to -50, and to write the arguments down the column from 0-00
to -50, and upwards from -50 to l-OO, attending to the sign of the second func-
tion ; and this is accordingly the arrangement in the table. Table II. (pp. 104,

105) contains < <^ - ^\ ^'i^'- ^\ and <^ " l'>(f^l±^ from
^ 2' 12 ' 24 ' 240

a; = 0-00 to iB = 1-00 at intervals of "01, the first to 5 and the others to 4

places. The first two have differences added.

[T. v.] (pp. 106-150). Natural sines, tangents, and secants throughout
the quadrant to every minute, to 5 places, without differences.

[T. VI.] (pp. 151-169). Table of squares to 10,000, arranged as in a
table of logarithms, the last figures of the squares (which must be 0, 1, 4, 5,
6 or 9) being printed once for all at the bottom of the columns.

The other tables are either astronomical or meteorological. There are 13 pp.
of formulsB.

Rankine, 1866. T. I. Squares, cubes, reciprocals (to 9 places) and five-
figure logarithms of numbers from 100 to 1000.

T. 1 A. Square and cube roots (to 7 places), and reciprocals (to 9 places) of
primes from 2 to 97.

T. 2. Squares and fifth powers of numbers from 10 to 99.

T. 2 A. Prime factors of numbers up to 256.

T. 3. Hyperbolic logarithms of numbers to 100, to 5 places,

T. 3 A. Ten multiples of the modulus and its reciprocal.

T. 4. Multipliers for the conversion of circvilar lengths and areas, viz. a
few multiples of tt and its reciprocal, square roots, &c.

T. 5. Circumferences and areas of circles, viz. ird (to 2 'places), and -2~

(to the nearest integer), from d = 101 to d = 1000.

T. 6. Arcs, sines, and tangents for every degree, to 5 places.

Raper, 1846. T. I. Six-figure logarithms of numbers from 1 to 100 and
from 1000 to 10,000, with proportional parts at the foot of the page.

T. II. Log sines for every second from 0° to 1° 30', to five places.

T. III. Log sines for every ten seconds from 1° 30' to 4° 31', to 6 places,
with proportional parts.

T. IV. Log sines, tangents, and secants for every half minute of the qua-
.drant, to 6 places, with proportional parts.

T. V. A page of constants.

Raper, 1857. T. 21 a. Logarithms for reducing daily variations, viz. log
1440"' _ log X, from a; = 1-" to a; = 1440" (= 24'') at intervals of a
minute, to 4 places, the arguments being expressed in hours and minutes.

T. 64. Six-figure logarithms of numbers to 100, and from 1000 to 10,000,
arranged as is usual in seven-figure tables, except that the logarithms are

ON MATHEMATICAL TABLES. 123

printed at full length ; tho proportional parts are given at the foot of the
page.

T. 65. Log sines, tangents, and secants to every quarter point, to six
places.

T. 66. Log sines of small arcs, viz. for each second to 1° 30', thence (T.
67) for every ten seconds to 4° 31', to 6 places, the logarithms being printed
at length ; T. 67 has proportional parts.

T. 68. Log sines, tangents, and secants (printed at full length) for every
half minute of the quadrant, to 6 places, with differences and proportional
parts for 1", 2". . . .30" (= half a minute) beyond 3°, semiquadrantally
arranged ; arguments also expressed in time.

T. 69. Log sin^ h from, x = toa;i= 180° at intervals of 15" (arguments

expressed also iu time), to 6 places ; all the logarithms printed at full length :
no differences.

T. 74. Proportional logarithms, viz. log 10800" — log .v from a; = 1 to
X = 10800" ( = 3° or 3'') to every second, the arguments being expressed in
degrees (or hours), minutes, and seconds, to 4 places ; the other tables are
nautical &c.

Resrnaud, 1818. Tho trigonometry occupies 182 pages ; and after the
diagrams are inserted Lalanbe's logarithms, vfhich are quite disconnected
from the work.

F' [T. I.] Five-figure logarithms to 10,000, arranged in columns, with cha-
racteristics and differences ; the number of degrees, minutes, &o. for the first
number in each column (viz. for every thirtieth number) is given at the top.

[T. II.] Log sines and tangents for every minute of the quadrant, to

5 places', with differences.

Riddle, 1824. T. IV. Log sines, tangents, and secants to every point
and quarter point of the compass, to 6 places.

T. V. Six-figure logarithms of numbers to 100, and from 1000 to 10,000,
with differences, arranged as usual.

T. VI. Log sines, tangents, and secants to every minute of the quadrant, to

6 places, with differences, semiquadrantally arranged. [The hBading of this
table in the book is inaccurate.]

T. XXVIII. Natural versed and suversed sines, viz. 1— cos» and 1 -(-cos a*,
for every minute of the quadrant, to 6 places, with proportional parts for
1", 2" . .. 60", so that the tabular results can be taken out very easily to
seconds. The extreme left- and right-hand columns serve both for minutes
in the arguments and for multiples in the proportional parts. The first
figure of the versed sine and the first tM'o of the suversed sine are generally
omitted throughout.

T. XXIX. Proportional logarithms, viz. log 10800" — log x from x =
to iT = 10800" (= 3° or 3"), the arguments expressed in degrees or hours,
minutes, and seconds at intervals of 1", to 4 places.

The book contains 34 tables, the rest of which are nautical. The navi-
gation &c. occupies 299 pages.

Rios, 1809. The first edition was published in 1806 ; and this is the
second. The tables are identical with those in the Spanish reprint of 1850
described below, so that the description of the. latter will suffice. The
numbers both of the tables and the pages are the same in both ; and the only
difference is that the headings of the tables &c. in the 1809 edition are in
English. A list of errors in this edition is given in the reprint of 1850.

Although the title of the Spanish reprint is given in the list in § 5, we have

124 KEPOBT— 1873.

thought it would bo more convenient to give the work the date of 1809, as
this more properly represents the time of appearance than does 1850.

T. XIV. Proportional logarithms for every second to 3°, to 5 places.
This table only differs from T. 74 of Eapek in there' being 5 instead of
4 places given.

T. XV. Pive-figure logarithms of numbers from 10 to 10,200, with the
corresponding degrees, minutes, and seconds. '

T. XVI. (pp. 382-472). Log sines, cosines, secants, cosecants, versed, co-
versed, suversed, and sucoversed from 0° to 45° at intervals of 15" (with
arguments also in time), to 5 places. The term " versed " (versos) is used
for semiversed sine for bre^•ity, and so for the others ; the table thus gives
log I (1 + cos .■») and log J (1 + sin x). The log sines, cosines, &c. are on
the left-hand pages, and the log versfed &c. on the right-hand pages. The
table, altered in arrangement so as to malie it quadrantal, is reproduced in
SiANSBUKT, 1822. There are also given some small tables to convert arc
into time, and vice versd, on p. 472.

These tables are all included under the heading ' Tablas logaritmicas y
tablas para convertir partes de oirculo en tiempo y viceversa.'

A list of errata in the London edition of 1809 is given at the beginning
of the edition of 1850.

Roe. T. I. Seven-figure logarithms of numbers from 1 to 100,000,
with characteristics unseparated from the mantissas. All the figures of the
number are given at the heads of the columns, except the last two, which
run down the extreme columns ; 1 ... 50 on the left hand, and 50 . . . 100 on
the right-hand side. The first four figures (counting the characteristics) are
printed at the top of the columns. There is thus an advance halfway to-
wards the modern arrangement, and the final step was made by John Newton
(1668). This is the first complete seven-figure table that was published; It
is formed from Vlacq by leaving out the last three figures, without increasing
the seventh when they are greater than 500.

T. II. Logarithmic sines and tangents for every hundredth part of a
degree (viz. -guVu part) of the quadrant, semiquadrantally arranged, to
10 places, with characteristics, which, however, are separated by a comma.

The work is very rare : the copy we have seen belongs to the Eoyal Society.

Riimker, 1844. T. I. Six-figure logarithms of numbers from 1000 to
10,000, arranged consecutively jn columns and divided into decades, with the
proportional parts for each decade by the side of it.

T. II. Log sines and tangents for every ten seconds to 2°, and log sines,
tangents, and secants for every minute from 0° to 45°, with differences, to
6 places ; the logarithms written at length.

T. III. Natural versed sines to every minute to 180°, with proportional
parts for the seconds, to 6 places.

T. rV. Logarithmen-Steigezdt, viz. log versed sines for every minute to 12'',
to 6 places, with differences for one second (corresponding to 0'' 0" ; the
table gives instead of — oo). «

T. XXIV.' Proportional logarithms for every second to 3°, to 4 places •
same as T. 74 of Eapeb.

In aU cases the logarithms are written at length. The other tables are
nautical.

^Salomon, 1827. This work we have not seen ; but as Eogg has given
a description of several of the tables, and we see no likelihood of meeting
with the book, we here give his account. There are 13 tables, of which
the most noteworthy are the following : —

ON MATHEMATICAL TABLES. 13S

T. I. Squares, cubes, square and cube roots (to bow many places is not
stated) of all numbers from 1 to 10,000 conveniently arranged.

T. II. Factors (except 2, 3, 5, and 11) of numbers from 1 to 102,011. •
T. YII. Six-flgure logaritbms of numbers to 10,800 (the last 800 to
7 places).

T. YIII. Briggian and hyperbolic logarithms of all numbers from 1 to
1000, and of primes from 1009 to 10,333, to 10 places.

T. IX. Logdrithmic canon for every second of the first two degrees, and
then. for every ten seconds of the rest of the quadrant (to 6 or 7 places, we
suppose).

T. XII. Natural sines and tangents for every minute, with differences. Eogg
adds that the printing and paper are good for Germany, but that he has made no
comparison to determine the correctness of the table ; the two pages of errata,
however, show (he remarks) that there was not so much care taken as with
Sherwin, Gaedinee, Cailei, HTTiioif , Taylok, or Veoa. Eogg's account is to
be found on pp. 254 and 399 of his ' Bibliotheea.' See also Gernerth's tract.
"^Schlbmilch [1865 ?]. Pive-figure logarithms to 10,909 ; table for the
conversion of Briggian into hyperbolic logarithms ; logarithms of constants ;
circular measure of degrees, minutes, and seconds ; natural functions for every
ten minutes of the quadrant ; log functions, for every minute ; reciprocals,
square and cube roots, and hyperbolic logarithms of numbers to 100 ; elliptic
quadrants ; physical and chemical constants.

The above description is taken from an advertisement.
Schmidt, 1821. [T. I.] Five-figure logarithms to 100, and from 1000
to 10,000, with proportional parts.

[T. II.] Log sines and tangents for every minute of the quadrant (semi-
ijuadrantally arranged), to 5 places, with differences.

[T. III.] Natural sines (to 5 places) and tangents (to 5 places when loss
than unity, above that to 6 figures) for every minute of the quadrant.

■[T. IV.] Circular arcs, viz. circular measure of 1°,'2°. . . 90°, 120° . . .
300°, 360°, of 1', 2' . . . 60', and of 1", 2" . . . 60", to 12 places.

[T. v.] Squares and cubes of all numbers from unity to 1000, with two
subsidiary tables to extend the table to 10,000; the latter are of double
entry, and contain :— (i) (2 a + c) c for c= 1, 2 . . . 9 and a=10, 11 . . . 99,
and i.o and 2 6 c for the same values of c and for 6 = 1, 2 ... 9 j and (ii)
(3 «'^ + Qac + c') c for c = 1, 2 ... 9, and a = 10, 11 . . . 99.

There are a few other small tables for the solution of triangles, refrac-
tions, &c.

Schrbn, 1860. T. I. Seven-figure logarithms to 1000, and from 10,000
to 108,000 (the last 8000 being to 8 places), with proportional parts to one
place of decimals, so that they are in fact multiples. The change in the line
is denoted by an asterisk prefixed to the fourth figure of all the logarithms
affected. The degrees, minutes, &c. corresponding to every number (regarded
as that number of seconds) in the left-hand column, and also corresponding
to these numbers divided by 10, are given. At the bottom of the page also S
and T (and also the log sine and tangent) are added for every 10" (§ 3,
art. 13, p. 54). "When the last figure has been increased there is a bar
subscript, which, being more obtrusive, is not so good as Babbage's point.
^The table is followed by the first 100 multiples of the modulus and its reci-
procal, to 10 places.

T. II. Log sines and tangents for every ten seconds of the quadrant, to
7 places, with very complete proportional-part tables (or more properly mul-
tiples of the differences). The increase of the last figure is noted as in T. I.
T. III. Interpolation table, viz. the first 100 multiples of aU numbers

126 REPORT— 1873.

from 40 to 410. The table occupies 75 pages ; and on eacli double page are
given the proportional parts to hundredths of 1, 2, 3, 4, and 5 (viz. the first
100 multiples divided by 100 and contracted to one decimal place). The
last page of the book is devoted to a table for the calculation of logarithms,
and contains common and hyperbolic logarithms of n, 1-On, l-OOn, &c., n

being any single digit (or in other words, of 1 + -^ from a; = 1 to a; :^ 9

and n = 1 to m = 10), to 16 places. The figures are beautifully clear, and
the paper very good. The tables are of their kind very complete indeed.

We have seen errata in this work advertised in difierent numbers of
Cfrunert's ' Archiv der Mathematik und Physik.' See Sohron, 1865, below.

Schrbn (London edition), 1865. De Morgan remarked that in England,
though there existed minute- and second-tables of trigonometrical functions,
there was no good ten-second table ; and on learning from the publishers
that an English edition of ScHEoif was contemplated, he oiTered to write a
short preface, as, accuracy being taken for granted, these appeared to him to
be the most powerful and best ten-second tables he had seen : his ofier, how-
ever, was accompanied by the condition that a careful examination should be
made by Mr. Farley, sufficient to judge of the accuracy of the work, and that
the result should be satisfactory. Mr. Farley accordingly, examined 24 pages
selected at hazard, wholly by diiferences and partly by comparison with
Callet ; and the pages were found to be totally free from error ; so that the
general accuracy of the tables was assured. They are printed from the
same plates as in the German edition described above ; and the tabular matter
in the two seems identical in all respects.

Schulze, 1778. [T. I.] Seven-figure logarithms to 1000, and from
10,000 to 101,000, with differences and proportional parts. The proportional
parts at the beginning of the table, which are very numerous, are printed on
a folding sheet.

. A page at the end of this table contains the first nine multiples of the
modulus and its reciprocal, to 48 places ; also e to 27 places, and its square,
cube. . . .to its 25th power, also its 30th and 60th powers, the number of
decimals decreasing as the integral portion increases. Log tt (hyperbolic and
Briggian) is also given.

[T. II.] "Wolfram's hyperbolic logarithms of numbers to 48 places. The
numbers run from unity to 2200 at intervals of unity, and thence to 10,009,
only not for all numbers ; " von 2200 bis 10,000 ist sio hingegen nur f iir die
Prim- imd ctwas stark componirte Zahlen bereohnet, well das Uebrige durch
leichtes Addiren kann gefunden werden " (Preface). De Morgan says " for
all numbers not divisible by a single digit;" but this is incorrect, as 2219,
2225, &e. are divisible by single digits, while 9809 (least factor 17), 9847
(least factor 47) do not occur. In fact, at first a great many composite
numbers are tabulated, and near the end very few, if any. All the primes,
however, seem to be given ; and by the aid of Wolfram's tables we may
regard all hyperbolic logarithms of numbers below 10,000 as known. Space
is left for six logarithms, which Wolfram had been prevented from computing
by a serious illness. These were supplied in the ' Berliner Jahrbuch,' 1783,
p. 191. Mr. Gray points out an error in Wolfram's tahle ; viz. in log 1409,

1666 .... should be .... 1696 .... (' Tables for the formation &c./ 1 865,

p. 38).

- On Wolfram, see § 3, art. 16.

[T. III.] Log sines and-tangents for every second from 0° to 2^, to seven
places : the sines are on the left-hand pages, the tangents on the right-hand ;
no differences.

ON MATHEMATICAL TABLES. 127

[T. IV.] Logistic logaritlims to every second to one degree, to four places.
The pages in [T. III.] and [T. IV.] are not numbered.

[T. v.] is the first table in the second volume. It contains : — natural sines,
tangents, and secants to seven places, with differences ; log sines and tangents
to seven places, with differences (from 0° to 4° the simple difference, and from
4° to 45° one sixth part of the difference, is given) j and Napierian (see § 'd, ■
art. 17) log sines and tangents to eiglit places, without differences; allffor
every ten seconds for the first four degrees, and thence for every minute to 45°.
The Napierian logarithms (see first page of Preface to. the second volume) are
taken from the ' Canon Mirificus ' of Napiee, augmented by Uesinus. The
arrangement of the table is not very convenient, but perhaps the best
possible.

[T. VI.] (pp. 262, 263). Eirst nine multiples of the sines of 1°, 2°, 3°
.... 90°. One or two constants are given on p. 264.

[T. Vn.] Circular measure of all angles from 1° to 360° at intervals of
1°. This is followed by similar tables for minutes from 1' to 60' at intervals
of 1', and for seconds from 1" to 60" at intervals of 1", all to 27 places.

[T. VIII.] Powers, as far as the eleventh, of decimal fractions from -0 to
1-00 at intervals of -01, to eight places.

[T. IX.] Squares of numbers to 1000.

[T. X.] Cubes of numbers to 1000.

[T. XI.] Square roots of numbers to 1000, to seven places.

[T. XII.] Cube roots of numbers to 1000, to seven places.

[T. XIII.] The first six binomial-theorem coefiioients, viz. x, ^' ~ ' , ....

~i o n~ > ^°^ ^ = '01 to i» = 1-00, at intervals of -01, to seven

places.

The other tables connect the height and velocity of falling bodies, and
contain specific gravities &c. A table on the last page is for the conversion
of minutes and seconds of arc into decimals of an hour.

A table headed Rationale Trigonometrie occupies pp. 308-311, and is very
interesting. It gives right-angled triangles whose sides are rational and
such that tan gw (w being one of the acute angles of the triangle) is
greater than Jg-. Such triangles (though not so called here) are often known
as Pythagorean. Those with sides 3, 4, and 5 ; and 5, 12, and 13 are the
best-known cases ; and 8, 15, and 17, 9, 40, and 41, 20, 21, and 29, &c. are
among the next in point of simplicity. This table contains 100 such tri-
angles ; but some occur twice. It gives in fact a table of integer values of
a, h, c, satisfying a'-\-li'=(?, subject to the condition mentioned above s
tan |w, expressed both as a vulgar fraction and as a decimal, is given, as also
are w and 90°— a>. For a larger table of the same kind, see Sang, 'Edinburgh
Transactions,' t. xxiii. p. 757, 1864. On the whole, this collection of tables
is very useful and valuable.

[Schumacher, 1822 ?]. T. V. Eive-figure logarithms of numbers for
every second to 10,800" (3°), arguments expressed in degrees, minutes, and
seconds.

T. VI. Log sines for every second to 3°, to five places. There is no name
at all on the table ; but it is assigned (and no doubt correctly) to Schumacher
in the Eoyal Society's Library; and De Morgan, speaking of "WAENSTOEr/s
ScHiTMACHBB (1845), says that the original publication was Altona, 1822 ;
but there was an earlier edition, we believe, at Copenhagen, in 1820.

Shanks, 1853. The bulk of this work _([T. I.] pp. 2-85) consists of the
values of the terms in Mr, Shanks's calculation of the value of tt by Maohin's

128 KEPoaT— 1873.

formula, 7r = 16 tan- 1 |-4tan-i J^. The terms in the expansion both of
tan -1 i and tan "' ^a? "'^s §'"'^°- separately to 530 places. The former

occupy 60 pp. and extend to „ g,„ ; and the latter occupy 24 pp. and ex-
tend to oiQ.oQQaia • "While the work was passing through the press Mr,

Shanks extended his value of ir to 607 decimals ; and to this number of
places it is given on pp. 86 and 87 of the book.

[T. II.] (pp. 90-95) gives every twelfth power of 2 (viz. 2", 2", &c.) as far
as 2^^' (which contains 212 figures).

On p. 89 are given the values of e, log, 2, log, 3, log, 5, and log, 10, to 137
places, and the modulus to 136. Yalues of these quantities were given also
by Mr. Shanks to 205 places (Proc. lloy. Soc. vol. vi. p. 397). The value of.«
was verified by the reporter to 137 places by calculation from a continued
fraction (see Brit. Assoc. Report, 1871, pp. 16-18, sectional proceedings).
The same writer also showed in vol. xix. p. 521 of the ' Proceedings of the
lloyal Society,' that Mr. Shanlcs's values of log 2, 3, 5, and 10 were inaccurate
after the 59th place (all owing to one error in a series on which they depended),
and deduced the correct values to 100 places. These results were verified by
Mr. Shanks, who has recalculated the values of these logarithms, as well as
that of the modulus, to 205 places : they are published in vol. xx. p. 27 of
the ' Proceedings of the Eoyal Society ' (1871).

Mr. Shanks's 607-place value is given in Knight's ' English Cyolop£edia,'
(Art. "Quadrature of the Circle") copied from the work under notice ; and it
has been verified by a subsequent calculation of Kichter to 500 places. A
list of the calculators of w, the number of places, &c. to which they have
extended their calculations, with references to the places where they arc
to be found, is given by Bierens de Haan on a page at the beginning of his
" Tables d'Integrales Definies " in t. iv. of the Amsterdam Transactions.
This page, however, does not appear in the separate copies of the tables
(the ' Nouvelles Tables/ Leyden, 1 867). For an extended and corrected copy
of this Hst, see ' Messenger of Mathematics,' December 1872, and some addi-
tional corrections in the same iTournal for July 1873 (t. iii. pp. 45, 46).

Some years ago Mr. Shanks calculated the reciprocal of the prime number
17389 so as to exhibit the complete circulating period, consisting of 17388
figures, and placed a copy of it in the Archives of the lloyal Society, Quite
recently he has extended his calculation of ?r to 707 decimal places (Proc.
Eoy. Soc. vol. xxi. p. 318). Mr. Shanks has sent us three corrections to this
paper ; viz. the 459th, 460th, and 461st decimals in ir should be 962 instead
of 834, and the 513th, 514th, and 515th decimals should be 065 instead of
193 ; also the 75th decimal of tan "^^ should be 8 instead of 7. The two
corrections in tt apply also to the work under notice.

Sharp, 1717, [T, I.] (p. 40). The first hundred multiples of It, to 21 places,

[T. II.] Areas of segments of circles. The area of the whole circle is
taken as unity; and the argument is the versed sine (or height of the
segment), the diameter being taken as unity. The table then gives areas to
17 places for arguments -0001 to -5000 at intervals of -0001, with differences.
Thus, strictly, the argument is the ratio of the height of the segment to the
diameter, and the tabular result the ratio of the area of the segment to that
pf the whole circle. The table occupies 50 pp., and is the largest of the kind
We have seen.

[T, m.] Tdbhfor computing the solidity of the upright hypeiiolic section
of a cone, viz. for facilitating the calculation of the volumes of segments of

ON MATHEMATICAL TABLES. 129

right circular cones, the segment being contained by the base of the cone (a
segment of a circle), a hyperbolic section perpendicular to the base, and the
curved surface. The use of the table (which contains 500 values of the
argument and occupies 5 pp.) is explained on pp. 24-26 of the work.

[T. IV.] Briggian logarithms of numbers from 1 to 100, and of primes
from 100 to 1100, to 61 places; also of numbers from 999,990 to 1,000,010,
to 63 places, these last having first, second .... tenth differences added. The
logarithms in this table were copied into the later editions of Shebwin and
other works.

The _ portion of the work which contains the tables is followed by a
" Concise treatise of Polyedra, or solid bodies of many bases " (pp. 32).

The work is universally attributed to Abraham Sharp, and no doubt exists
as to his having been the author.

[Sheepshanks, 1844.] [T. I.] Four-figure logarithms from 100 to
1000, arranged as in seven-figure tables, with proportional parts.

[T. II.] Log sines and cosines (the arguments being expressed in time) to
24'' at intervals of 1", to four places, with proportional parts for multiples of
10" (to 60'). Also log sines to 1'' for every 10', with differences for 1'.

[T. Ill,] Log sines, cosines, tangents, and secants from 0° to 6° at
intervals of 1', thence to 84° at intervals of 10', and then at intervals of 1' to
90°, to four places. In the parts of the table where the intervals are 10',
differences for 1' are given.

[T. lY.] Natural secants and tangents from 0° to 80" at intervals of 10',
with differences for 1', and then to 86° at intervals of 1', with diflferences for
10", to four places.

[■r. v.] Modified Gaussian logarithms. There are two tables. The first

gives log j 1 -f - I as tabular result for argument log x, the range of log *

being from -000 to -909 at intervals of -001, from -90 to 2-00 at intervals of

•Gl, and thence to 4-0 at intervals of -l. The second table gives log 1 1 j

as tabular result, corresponding to the argument log ,r, the range being from
•000 to 1-000 at intervals of -001, from 1-00 to 3-00 at intervals of -01, and
from 3-0 to 6-0 at intervals of -1 : both tables to four places, with propor-
tional parts.

[T. VI.] Log sin' (i hour angle) from 0* to 9'' at intervals of 1'", to four
places, with proportional parts for multiples of 10' (from Eipbk).

[T. Vll.] Antilogarithms, for logarithms from -000 to 1-000 at intervals
of -001, to four places, with proportional parts.

There are also two or three astronomical tables.

De Morgan states that the work was issued under the title given in § 5 in'
1846, and two years previously without name or titlepage. It is from one of
these earlier copies that the above description has been written ; ve have
s.een no copy bearing either author's name or date.

Sherwin, 1741. [T. I.] (which follows p. 35 of the introduction) gives
Briggian logarithms to 61 places of all numbers to 99, and the logarithms of
primes from 100 to 1097, calculated by Abraham Sharp (see Shabp, 1717,
[T, IV.]).

[T; II.] Briggian logarithms of thirty-five other numbers (viz. 999,981
—1,000,015), to 61 places, with first, second, third, mi fourth differences,'
to 30 places (Shabp [T. IV.]).

[T. III.] Seven-figure logarithms of numbers to 1000, and from 10,000
1873. K

130 SEPOET— 1873,

to 101,000, with proportional parts. Tho proportional parts near tho begin-
ning of tUe table, being too voluminous for insertion on tho page, are printed
on a fly-sheet, and bound up facing the introductory page of the table.

[T. lY.] Natural and log sines, tangents, and secants for every minute, to
seven places. Differenoea for the logarithmic functions are added, but not
for the natural ones.

[T. Y.] Natural and log versed sines from 0° to 90° at intervals of o
minute, to seven places. Port of a page at the end of [T. V.] is occupied by
a small table to convert sexagesimals into decimals, &c., and vice versd.

The' remaining table (of difference of latitude and departure) is not in-
cluded in this Report (see § 2, art. 12).

Sherwin wont through five editions ; but as none were stereotyped, some of
the later are less accurate than the earlier. De Morgan remarks, " Second
edition, 1717 ; third revised by Gardiner, and the best, 1742 ; fifth and last,
1771, very erroneous — the most inaccurate table Hutton ever met with."
In speaking of the third edition we at first thought that De Morgan should
probably have written 1741 instead of 1742, as the edition we have described
bears the former date, but we have since seen a copy of 1742.

We possess an edition (1726) which contains a list of " Errata for the
second edition of Sherwin's Mathematical Tables " by Gardiner. In this edi.
tion, in place of [T. I.] and [T. II.] there are given two pages (pp. 28 and 29)
headed " M. Brigg's {sic) Logarithms for all Numbers, from 1 to 100, and for
all Prime Numbers from 100 to 200, calculated by that Ingenious Gentleman
and Indefatigable Mathematician, Mr. Abr. Sharp, at Little Horton, near
Bradford in Yorkshire." The logarithms are given to from 50 to 60 places
(not all to the same esrtent).

"We have also before us an edition of 1706 ; and the dedication, which is
the same in all the editions we have seen, is dated July 12, 1705. The table
on pp. 27 and 28 is the same as in the edition of 1726 ; but at the end of the
introduction is a table of errata, which are corrected in this latter edition.
Tho titlepage of the editions of 1705, 1706, and 1726, and perhaps other
dates, runs, " Mathematical Tables. . . .with their Construction and Use by
Mr. Briggs, Mr. WaUis, Mr. HaUey, Savihan Professors of Geometry in the
University of Oxford, Mr. Abr. Sharp" (the names of the authors being
placed one under the other); and in the edition of 1706 is added, "The
whole being more correct and complete than any Tables extant." Sherwin's
name does not, therefore, occur on the titlepage at all ; but the preface is
signed and the tables were prepared by him, so that the work is universally
known as " Sherwin's Tables." In library patalogues, however, it will gene-
rally be found entered under the name of Briggs, Wallis, Halley, or Sharp.

In the edition of 1741, the names of Briggs, Wallis, Halley, and Sharp do
not appear on the titlepage, but we have " The third edition, carefully
revised and corrected by "WiUiam Gardiner " instead.

It will be seen that there is some confusion in the editions, as, if Do
Morgan is correct in saying that the second edition was published in 1717,
the edition of 1726 would be the third, and that of 1741 the fourth.

The Royal Society's Library contains a copy with " 1705 " on tho title-
page, while the edition of. 1706 (which is in the library of Trinity College,
Cambridge) has the date printed in Roman characters, MDCCYI.

We have seen (in the Graves Library) the fourth edition, 1761 j and the
British Museum contains, besides the editions of 1717 and 1742, the fifth
edition, " revised and improved by S. Clark " (1772), while the Cambridge
University Library has the same edition with the date 1771.

ON HATIIEMATICAL TABLES. 131

Thfe edition? we have seen are 1705 and 17G6, 1717,- 1726; the third
edition 1741 and 1742, the fourth 1761, and the fifth 1771 and 1772. It
thug appears that it -(fas not at all an uncommon thing (probably, as the
impression was being ijiade up from time to time) to advance the date by one
year. The first four dates we may distribute among the first two, editions as
we please j most likely 1705, 1706, and 1717 for the first, and4726 for the
geoond.

Eogg (p. 401) gives the editions as 1706, 1742, 1763, and 1771 ; but else-
where (p. 262) be speaks qf the fifth as of 1785,' which must be iiioorrect.

De Haau (• lets over Logarithmentafels,' p. 57) gives the dates of the
editions as 1708, 1717, 1726, second 1742, 1751, 1763, fifth 1771. The
subject of the dates of the editions of Sherwin is discussed at some length in
the ' Monthly Notices of the Royal Astronomical Society ' for March and
May 1878 (vol. xxxiii. pp. 344, 454, 455, 457). Mr. Lewis, in bis letter
to the reporter, printed in the second of these papers, mentions 1717, 1742,
1761, and 1771 as the dates -of the editions he had seen, agreeing perfectly
with those mentioned by De Morgan, Lalande ('JBibljog, Astron.'), and the
results of our own observation, He remarks that Barlow gives 1704 and
CaUet 1724 as dates of editions, of which the former may be dismissed at
once as an obvious blupder. The editions therefore that we have not seen,
but which piay exist, are those of 1724, 1751, and 1763. About any of
these or any others we should be glad to receive information,

Eogg mentions that Shbeww has often been confounded with Gabdiifee,
even by Kiistner and Bugge,

With regard to the accuracy of the tables, Huttoit writes (we quote, from
p. 40 of the Introduction to his tables, 3rd edit. 1801) ;— " The first edition
was in 1706 j b^^t the third edition, in 1742, which was revised by Qardiiier,
is esteemed the most correct of any, though containing many thousands of
errors in the final figures : as to the last or fifth edition, in 1771, it is so erro-
neously printed that no dependence can be plaeed in it, being the most in-,
accurate book of tables I ever Imew ; I have a list of several thousand errors
which I have corrected in it, as well as in Gardiiier's octavo edition."

i)e Haan ('lets' &c., p. 26), speaking of the 1742 edition, gays that it-
contains the logarithms of the numbers from 999,980 to 1,000,020 to 61
places ; but on examination we find that the above description of [T. II.]. ia
correct. The advertisement to the book itself is no doubt the source of the
error ; for it is there said to contain the logarithms of the 41 numbers from
999,980 to 1,000,020, whereas it really contains the logarithms of the 35
. pumbers from 999,981 to 1,000,015. '

Sherwia's tables are of historical interest as forming part of the main line
of descent from Bbiggs ; and the diflferent editions cover the greater part of
the last century. The chief sucoossion (considering only logarithms of num-
bers) is Bkigos, VLAca, Eoe, John Newion , Sheewdst, Gaedineb ; and then
there are two brS.nches, viz. Huitoir founded on Shebwin, and Callet on
Gaedineb, the editions of Vesa forming an offshoot.

Shortrecle (Compendious logarithmic tables), 1844, Small tables of
common logarithms with sexagesimal arguments, logarithms to 12,600, anti.^
logarithms from to -999, log sines and tangents to 5', also from 0° to 3°,
and from 3° to 5° for every two minutes ; aU to five or six places. Tha
tract only contains 10 pp.

Sbortrede (Tables), 1844. T, I. Seven-figure logarithms to 10,800 with
characteriatios, but without differences, and from 10,800 to 120,000,- with
differences, and their first nine multiples at the bottom of the page : the num-

K 2

132 REPORT— 1873.

ber of Sogrees, minutes, and seconds corresponding to the numbers in the
number-column multiplied by 10 is given throughout ; and at the top of every
page arc printed, to seven places, the logarithms of certain constants, viz.
of 360°, 180°, 90°, 1°, 24*, 12", 3*, 1", and radius (all expressed in seconds)
of arc 1", TT and M the modulus. The change of figure in the Ime is
denoted by a " nokta," the same as that employed subsequently by Mr. Sang
(see Sang, § 3, art. 13) ; and its use is open to the same objections here as
■there.

T. II. Antilogarithms, viz. numbers to logarithms from -00000 to l-OOOOO
at intervals of -00001, to 7 places, with differences and multiples at the
bottom of the page. The same logarithms of constants are given on the top
of the page as in T. I. ; and the change in the line is denoted in the same
way. At the end of this table (p. 195), under the head " Useful Numbers,"
the logarithms of some constants are given.

T. III. (pp. 598). Log sines and tangents to every second of tJie circum,'
ference, to 7 places (semiquadrantally arranged), the arguments throughout
being also given in time. The use of the vrord circumference instead of
quadrant in this description is justified by the fact that the signs are given
for the different quadrants at the top and bottom of the page : thus we have on
the first page, at the top, 0° Sin +, 90° Cos—, 180° Sin — , 270° Cos +, and
at the bottom 89? Cos +, 179° Sin +, 269° Cos -, 359° Cos — , and the same
for the tangent and cotangent, the arguments being also expressed in
time. Complete proportional parts are given throughout for tenths of a
second of space, and for the first six hundredths of a second of time, both
for the sine and tangent ; but near the beginning of the tables coefficients of
correction for first and (sometimes) second differences are added instead. The
arguments, as before stated, are given also in time ; so that cortesponding to
1", 2", .3", &c. we have -06», -13', -20', &o. This tabic is the most complete of
the kind we know of, and is unique ; the figures are clear ; and the objection
to the "nokta" does not apply here; in one column (p. 142) there are two
changes on the page.

T. V. Seven-place log sines, tangents, and secants to every point and
quarter point of the compass.

T. XXXVIII. Lengths of circular arcs, viz. circular measure of 1°, 2°, 3°
.... 180°, of 1', 2', . . . . 60', of 1", 2", .... 60", and of 1'", 2'", .... 60'", to 7
places.

T. XXXIX. Proportional parts to hundredths of the reciprocal of the
modulus, viz. 2-302 . . ., to 8 places.

There are thirty-nine tables in the book (T. XLI. is the last ; but XXXV.
and XXXVI. are accidentally omitted), the others being astronomical or me-
teorological &c.

The paging recommences withT. III. and proceeds to p. 634. See Ssoni
KEDE, 1849 (next below).

Shortrede, 1849. This is a second edition of the work of 1844, and is
in 2 vols. There is a preface of xxv pages to vol. i. T. I. and II. are the
same as T. I. and II. in the 1844 edition; T. III. is a small ten-
place table of the lengths of circular arcs. T. IV. and V. are for finding
logarithms and antilogarithms to many jjlaces ; viz. colog (1 + -Olm)
.. .colog (1 + -01' n), (fee. are given for n. = 1, 2,.. .100, to 16 places, and
colog (1 + -01 «).. .colog (1 + -01" n) for m = 1, 2,. .,10, to 25 places
(initial ciphers being omitted). There are added small auxiliary tables
for facilitating the resolution of numbers into convenient factors. T.
VI. The first hundred multiples of the modulus and its reciprocal to 32

ON MATHEMATICAL TABLES. 133

places. T. VII. (which occupies six closely printed pages). Modified Gaus*
sian logarithms. B (=log T+i) and C ^=log _^^ are tabulated for argu-
ment A (=log x), to 5 places, from A=5 to 3 at intervals of -1 ; from A = 3
to 2-7 at intervals of -01; from A = 27 to 1-3 at intervals of -001 ; and
from A=l-3 to 3-0 at intervals of -01, and thence to A=6 at intervals of -1,
T. \III, Log (1 . 2 . 3. .a;) from a;=l to a;=1000, to 5 and (for the argu-
ments ending in 0) to 8 places. '

Then follow 2 or 3 pp. of barometric &c. tables, and a page of constants

(including a small table of log EL^^ and the same for the tangent). .

. The second volume contains T. III. of the 1844 edition, followed by some
spherical-trigonometry formulae, and the same page of constants as in vol. i.

In the advertisement to the second (1849) edition, Shortrede says "a'
small edition of this work was published in 1844, before I had an opportu-
nity of seeing it complete, which in several respects was such as I did not
like. In the present edition many alterations have been made to conform it
more to my views; and for the convenience of purchasers it is now published
in two separate volumes." The prices of the two volumes are, Vol.I.12s., and
Vol. II. 30s. ; it is worth noting this, as we have seen it stated that the price
of Shortrede's logarithms (by which some might understand the wholo work)
is 12s. De Morgan says, " They [Shortrede's tables] first appeared in
1844; but some defects and errors having been found, the edition of 1844
was cancelled, and a new edition from corrected plates issued in 1849."
This may be true; but although we have seen four copies of the 1844 edi-
tion in different libraries, we were not able to obtain a sight of the 1849
edition anywhere till we bought it. Our copy of Vol. i. is dated 1849, and of
Vol. ii. 1858. There are few tables in which, relatively to the number of
figures, the pages are so clear, and the logarithmic canon to seconds is much-
the most complete we have seen. Every one must agree with De Morgan
that the work shows extraordinary energy and public spirit. This is the
most complete second canon in existence, and is the most accessible. Only
two others have been published : — Michael Taxlok, 1792, which has several
defects attending its use ; and Baoat, 1829, which is scarce.

A list of twenty-six errors (nearly all in the antilogarithms) is given by-
Shortrede himself in the ' Monthly Notices of the Eoyal Astronomical
Society ' for January 1864 ; and a supplemental list is added in the same
publication for May 1867, where he says that "the unauthorized issue in
1844 contains several others." One erratum is also given in the 'Monthly
Notice ' for April 1867. Shortrede adds that the great majority of the
errata were communicated to him by Mr. Peter Gray.

In the 'Insurance Eecord' Mr. Filipowski charged Shortrede with having
corrected his table by the aid of his (Filipowski's). That the charge was
utterly unfounded is proved by the letter of Mr, Peter Gray (' Insurance
Eecord,' June 9, 1871), who states that the errata in Dodson were given t(»
Shortrede by himself (Mr. Gray) ; and we have seen reason to impute un-
fairness to Mr. Filipowski in another matter with regard to Dodson (so-
Filipowski, 1849, § 4). Mr. Gray has kindly placed at our disposal his
copious list of errors in Dodson, of which we hope to make use in a sub-
sequent Eeport.

Shortrede did not pay sufficient attention to the examination of the errata-
lists of previous works ; and, in consequence, his tables contain a much greater
number of the hereditary errors that had descended from Vlacq than do the

134 iiEPOiii— 1873.

beat contemporary works. These errors are InBignificant in thcmBelves, ex-
cept in so far as ttey show the acquaintance of the author of a table with
the works of his predecessors. Shortrede was absent in India duiing the
publication of the 1844 edition (which contains seven of these errors) ; but
that of 1849 was published under his own superintendence, and still it con-
tains six, while Babbaoe, HIjlsse's Vesa, and other works of earlier date
have but one. See 'Monthly Notices of the Roy. Ast. Soo.,' March 1873,
t. xxxiii. p. 335 ; and Gernerth's tract (§ 3, art. 13, p. 55).

Stansbury, 1822. [T. I.] Small table to convert arc into time.

[T. II.] Proportional logarithms for every second to 3°, to 4 places. Same
as T. 74 of Eapbb.

T. D. Log semitangents, viz. log — 2^&oma;=0 t0ir=180°atintorvals

of 15', to 3 places. This table occupies one page.

T. G. Proportional logarithms for every minute to 24\ viz. log 1440
—log X, the arguments being expressed in hours and minutes (and also in
arc), to 4 places.

T. H. (pp. 215-304). Log sines and secants, also log versed and sucoversed,

from 0° to 90° at intervals of 1 5" (arguments also expressed in time) , to 5 places .

By "versed" and "sucoversed" are meant " semiversed sine "and "semisu-

coversed sine " (the terms introduced by De Mendoza y fiios being used for

1 + cos X It 1 + sin X
brevity, see Eios, 1809); so that the table gives log ^ — and log ^ — .

This table was copied from T. XVI. of Eios j but there is a difference of
arrangement, as the original table gave log sines, cosines, &c., the arrange-
ment being semiquadrantal, while in the present work it is quadrantal,

T. X. Five-figure logarithms from 1000 to 10,000 ; no differences.

T. Y. Halves of natural sines, viz, | sin x from x=(f to a; =90° at in-
tervals of a minute, to 5 places, with proportional parts for seconds.

The other tables are nautical.

Stegmann, 1855. T. I. Six-figure logarithms to 119, and five-figure
logarithms, with differences, from 1000 to 10,000.

T. II. Antilogarithms from -0000 to -9999, to 5 places. A few tables of
atomic weights &c. are added. As in Filipowski's tables, the terminal 5 is
replaced by the Roman V when it has been increased.

The preface to these tables is signed by Stegmann, but his name does not
appear on the titlepage.

^Stegmann. This work we have not seen. Three errata in it are given
by Prof. Wackerbarth in the ' Monthly Notices of the Royal Astronomical
Society ' for April 1867 : and this is the only place in which we have seen
the table referred to. It is very possibly a five-figure hyperbolic logarithmio
table, similar to the same author's table of common logarithms just de-
scribed.

Janet Taylor, 1833. T. XYII. Log sines, tangents, and secants to
every quarter point, to 6 places.

T. XVm. Six-figure logarithms of numbers to 10,000.

T. XIX. Log sines and tangents for every 10" to 2°, and log sines, tan-
gents, and secants for every minute of the quadrant, to 6 places, with dif-
ferences.

T. XX. Natural sines for every minute of the quadrant, to 6 places.

T. XXI. Log versed sines to S"* at intervals of 5', to 5 places.

T. XXXVI. Proportional logarithms for every second to 3°, to 4 places j
same as T, 74 of Rapeb,

ON MATHJ5MATI0AL TABLES. 133

At tho end of the preface Mrs. Taylor makes the following curious re-
mark :—« Some errors have crept into the calculations from the multiplicity
oi entries &c. ; these, I trust, will claim the indulgence of the public ; for
the system on which I have worked being mathematically correct, and
founded on sound principles, any slight oversight in the figures can be of
but little moment, and very easily rectified." It is to be presumed that this
does not refer to the tables included in this Eeport, as they would not have
been calculated afresh.

Mrs. Taylor was also the author of a work on navigation, the tables in
which are described below.

Janet Taylor, 1843. T. 3. Log sines, tangents, and secants to evel-y
quarter point, to 6 places.

T. 4. Six-figure logarittinS of numbers to 10,000.

T. 5. Log sines and tangents for every 10" to 2° ; and log sines, tangents^
and secants for every minute of the quadrant, to e places, with diffefeiices,
T. 30. Log versed sines for every 5' to 8\ to 5 places.
T. 32. Ifatural sines for every minute of the quadrant, to 6 places.
T. 35. Proportional logarithms for every second to 3°, to 4 places : Same
as T. 74 of Kapbe.

Mrs. Taylor, as we learn from an advertisement, kept a nautical academy
in the Minories.

Michael Taylor, 1792. [T. I.] Logcirithms of numbers to 1200, to 7
places.

[T. IL] Logarithms of numbers from 10,000 to 101,000, to 7 placesi, with
differences and proportional parts. The change in the tliird figure, in tho
middle of the lino is not marked.

[T. III.] Table of log sines and tangents to every second of the quadrant,
to 7 places (semiquadrantally arranged). The change in the leading figures,
when it occurs in the middle of the column, is not marked at aU ; and it
requires very great care in using the table to prevent errors from this
cause. If any one is likely to have to make much use of the table, it will
be worth his while to go through the whole of it, and fill in with ink the first
after the change (making it a black circle such as is used to denote full
moon in almanacs), aUd also to make some mark that will catch the eye at
the top of every column containing a change. This ■vt'ill be a work of con-
siderable labour, but is absolutely necessary to ensure accuracy, It is no
doubt chiefly on account of the absence of any mark at a change that
Bagat has so completely superseded this table, though difference of size &c,
are also in favour of the former;

[T. I.] and [T. II.] present no novelty ; but [T. III.] is an enormous table,
containing about 450 pages, with an average number of about 7750 figures
to a page, so that it contains nearly three millions and a half of figures^
The left-hand pages contain sines and cosines, the right-hand tangents and
cotangents. This is unfortunate, as the sines and cosines (which are Used
far more frequently than the tangents and cotangents) are thus separated
at least a foot from the computer's paper as he works Avith the table on his
left; audit is well known that the number of errors of transcription is
proportional to the distance the eye has to carry the numbers. [T. IIIi] was
calculated by interpolation from Vlacq's ' Trigonometria Artificialis,' to 10
places, and then contracted to 7 ; so that the last figure should always be
correct. Taylor was a computer in the Nautical Almanac Ofiiee ; he unfor-
tunately died almost at the moment of the completion of his work, only five
pages remaining unfinished in the press at the time of Ms death. These.

136 REroKi'— 1873.

were examined; and the introduction &c. written, by Maskelyne. Some
errata, found among Taylor's papers, are given on p. 64 of the work ; and a
list of nineteen errata signed by Pond is published iu the ' Nautical Almanac'
for 1833. To this list is appended the remark :— " The above errata were
detected by collating Ta3'lor's Logarithms with the French manuscript tables,
now the property of C. Babbage, Esq. The arrangement for this examina-
tion was made by the late lamented Dr. Young ; a few days only before his
death he gave directions for its completion. — J. Pour."

"We do not know any thing further with regard to this examination, though
the fact that certain errors were found in Taylor by comparison with the
French tables is well known ; but there must be some mistake, as the French
tables could not have been even temporarily the property of Babbage. In
the preface to his tables, Babbagb states that while on a visit to Paris he
availed himself of the opportunity of consulting the great manuscript tables
preserved at the Observatory, and that he " enjoyed every facility for making
the comparisons which were requisite for this purpose [the preparation of his
seven-figure table], as well as making extracts necessary to me for other
calculations."

Bagay intimates £n his preface that he had found 76 errors in Taylor.

Taylor was also the author of the Sexagesimal Table (§ 3, art. 9) ; and wo
cannot but admire the undaunted perseverance that could enable him to com-
plete such monuments of industry in addition to his routine work as computer
in a laborious office.

Thomson, 1852. T. I. One-page table to convert arc into time.

T. X. Logarithms for finding the correction of the sun's declination &c.,
viz. log 1440— log X, from a;=l toiB=1440, to 4 pla:ces.

T. XI. Logarithms of the latitude and jpolar distance, viz. log secants to
every minute of the quadrant, to 5 places, without differences ; quadrantally
arranged.

T. Xll. Logarithms of the half sum and difference, viz. log sines and
cosines to every minute of the quadrant, to 5 places, without differences ; qua-
drantally arranged.

T. XIII. Logarithms of the apparent time or horary angle, viz. 2 log sin -^

from A'^O' to x=&' at intervals of 10% with proportional parts for seconds^
to 5 places,

, T. XV. Logarithms of the apparent altitudes, viz. log cosec ce — "5400,
from x=Q° to x= 89°, at intervals of a minute, to 4 places.

T. XVI. Logarithms of the apparent distatf.ce, viz, log sines and tangents
for every minute, froln 18° to 90°, to 4 places.

, T. XIX. Four-place proportional logarithms for every second to 3° ; same
as T. 74 of Eapbb.

T. XXIII. Logarithms of the sum and difference, viz. log sin g, from

^•=0° to .«=180°, at intervals of a minute, to 6 places.

T. XXIV. Six-figure logarithms of numbei-s from 1000 to 10,000, with
differences and tables for interpolating at the foot of the page. In this book
it is only required to find numbers corresponding to logarithms ; and the
tables are constructed with this view. There are given, therefore, the usual
differences (called first differences'), and the approximate results of the divi-
sion of 1, 2, 3, ... . 10, and ten or more higher numbers by them. By the second
difference is meant the difference between the given logarithm and the logarithm
next below it in the table.

Ojr MATHEMATICAL TABLES. 137

T. XXV. Natural versed sines for every minute to 120°, to 6 places, with
proportional parts for seconds.

The other tables are nautLcal &c.

Trotter, 1841. [T. I.] Six-figure logarithms of numbers to 10,000,
'with differences. This is followed by a small table to convert Briggian into
liyperTjolic logarithms (fee.

^[T. II.] Log sines, tangents, and secants to every quarter point, to 6
places.

, [T. III.] Log sines and tangents for every fifth minute of the quadrant,
to 6 places.

[T. IV.] Natural sines and tangents for every fifth minute of the quadrant,
to 6 places.

[T. v.] Areas of circular segments, to 6 places ; same as T. XIII. of
Hantschl.

[T. VI.] Squares, cubes, square and cube roots (to 6 places) for numbers
from 1 to 1000.

[T. VII.] Circular measure of 1°, 2°, . . . . 180°, of 1', . . , . 60', of 1", .... 60",'
and of 1'", 60'", to 7 places.

[T. VIII.] Eeciprocals of numbers from 1 to 500, to 9 places.

[T. IX.] Logarithms of numbers from 1000 to 1100, to 7 places.

[T. X.] Lengths of sides of inscribed and circumscribed polygons (up to a
20-sided figure), the diameter of the circle being unity, to 7 places.

[T. XI.] Hyperbolic logarithms of numbers from 1 to 100, to 8 places.

[T. XII.] For finding the^reas of oblong and oblate spheroids. A few
constants are given. The other tables are astronomical, meteorological, &c.
Some trigonometry &c. is prefixed at the beginning (pp. 102).

Turkish Logarithms &c. [1834]. The book commences on the last
page ; and the first table gives seven-figure logarithms of numbers from unity
to 10,080, arranged consecutively in columns, there being three columns of
arguments and tabular results to the page. Th6 tables begin at the last page,
as before remarked, the extreme right-hand column being the first column of
arguments ; to the left of it is the correspoiiding column of tabular results,
-%h6nto the left of that the second column of arguments, and so on. The
table occupies 84 pp. (up to p. 85). Then " follows " a table of log sines and
tangents for every minute of the quadra,ut (semiquadrantally arranged), the
sines and cosines being side by side, and separated by some "white" from
the tangents and cotangents. This table occupies 90 pp., and is followed by
a similar table of natural sines and tangents (to 7 places), which also occupies
90 pp. Except that the table runs in the wrong direction, it only difi'ers from'
an ordinary table in the ten digits being denoted by difi'erent marks from
those to which we are accustomed. .A few minutes' practice, however, is quite
sufficient to get used to the new numerals ; and then the table could be used
as wei as any other. There is no introductory or explanatory matter. The.
book is in the British Sluseum ; and the place and date in § 5 are taken from"
the Catalogue of the Library.

Ursinus, 1827. [T. I.] Six-figure logarithms to 1000, and from 10,000
to 100 000, without differences ; the values of S and T for finding log sines
and tangents of angles below 2° 46' 40" (see § 3, art. 13) are given at the top

of the page,

rT. li.] Log sines and tangents for every 10 seconds throughout the
quadrant, with differences, to 6 places.

TT III.] Longitudes of circular arcs, viz. circular measure of 1°, 2°, 3°,

360° of 1' 2', ... 60', and of 1", 2", 60", to 7 places. These are followed:

138 14EP011T— 1873. .

by a page giving tho sines of 3°, 6°, 9°, . . . , 87° accurately (». e, expressed as
radicals).

[T, IV.] Longitudes of chords, viz. lengths 'of chords subtending given
angles (the arguments) at the c"entre. The arguments proceed from 0° to^
108°, at intervals often minutes, and thence to 180° at intervals of 1°; and
the tabular results are given to 3 places.

[T. v.] Abacus trigonometrkus, viz. natural sines, tangents, and secants,
and log sines and tangents from 0° to 90° (quadrantaUy arranged), to
every ten minutes, to 6 placeS. Then foUow a few formula) and con-
stants.

Vega (Thesaurus, fol. 1794). T. I. (Magnus Canon logarithmerUm
vulgarium). Logarithms of numbers from 1 to 1000, without differences, and
from 10,000 to 100,999, with differences, to 10 places, arranged like an
ordinary seven-figure table. Proportional parts are also given, but only for
tho first two or three figures of the diflference. The table can thus be used
as an ordinary seven-figure table. A change in the fourth figure in tho
middle of the line is denoted by an asterisk prefixed to aU the logarithms
affected. T. I. occupies pp. 1-310. The last page and a half are devoted to
multiples of the modulus, a few constants, and a table to convert degrees (1°
to 360°) and minutes (1' to 60') into seconds.

T, II. (Magnus Canon logarithmorum vulgarium trigonometricus), Log
sines, cosines, tangents, and cotangents, from 0° to 2° at intervals of one
second, to 10 places, without differences, and for the rest of the quadrant at
intervals of ten seconds, also to 10 places, withCifferences. All this occupies
pp. 311-629, and is followed by 3 pp. containing natural sines for angles less
than twelve minutes, to every second, to 12 places.

The appendix occupies pp. 633-685 : p. 633 contains formute ; and pp. 634
and 635 are occupied with tables of the longitudes of circular arcs &c. Of these
tlie first gives the circular measure of 1°, ^, 3°, . . . .360°, the second of 1', 2',

3' 60', the third of 1", 2", 3" 60", all to 1 1 places ; the fourth is a

small table to express minutes and seconds as fractions of a degree. Pp. 636-^
640 are occupied with formulae for the solution of triangles ; and on pp. 641-
684 [T. m.] we have "Wolfram's great table of hyperboUc logarithms (see
SciruLZB, § 4). The sis omitted in ScHtrLZE are given ; and it is stated in the
preface that several errors have been corrected. The error pointed out by Mr.
Gray (see Schulzb [T. II.]) is reproduced. An error in log, 1099 is pointed
out by Prof. Wackerbarth in the ' Monthly Notices of the Boyal Astronomioal
Socie^' for April 1867.

Some of the errata found in Vlacq are indicated in the preface. These are,
as a rule, corrected in the book ; others, given in a list at the end of the in-
troduction, were found after the printing, and must be corrected in manu-
script before use. There is a third list at the end of the work (p. 685) ; but
it is identical with that at the end of the introduction.

In some copies the list at the end of the introduction is much more com-
plete than in others, the errors in VLiCQ being marked by an asterisk, and the
errata being also given in Latin and German. It is probable that additional
errata were found before the edition was aU made up, and that the original
list was suppressed and the new one substituted. In all copies the titlepago
is the same. See ' Monthly Notices of the Eoy. Ast. Soc./ June 1872, and
May 1873 (p. 454).

There is a great difference in the appearance of different copies of the work.
In some the tables are beautifully printed on thick white paper, with wide
margin, so that the book forms one of the handsomest collections of tables we

ON MATHEMATICAL TABLES. 139

havo Seen ; while in others the paper is thin and disooloured ; all are printed
from the same type.

The arrangement of T. I. (though about half the space that would be required
ii the logarithms and differences were written at length is thereby saved) is not
nearly so convenient as in VtAca j 1628, for there is danger of taking out a
wrong difference. Vega took great pains to free his tables of logarithms of num.
bers from error ; and he detected all the hereditary errors that had descended
from ViACft which affected the first seven figures of the logarithms. But as
several of these errors were corrected in his errata-list and not in the text, his
successors, who failed to study these lists sufficiently, were really less accurate
than he was. The last thousand logarithms that appear for the first time in
this Work were calculated by Lieut. Dorfmund at Vega's instigation.

T. II. is not reprinted entirely from Vlacq's ' Trigonometria Artificialis,'
as the logarithms for every second of the first two degrees were calculated for
the Work by Lieut. Dorfmund. Vega seems not to have bestowed on the tri-
gonometrical canon any thing approaching to the care he devoted to the loga-
rithms of numbers, as Gauss estimates the number of last-figure errors at from
31,983 to 47,746 (most of them only amounting to a unit, but some to as
much as 3 or even 4).

Vega offered a reward of a ducat for every error found in his table ; and
it is to be inferred from his preface that he intended to regard inaccuracies of
a unit as such, so that it was fortunate that no contemporary of his made an
examination similar to Gauss's. The paper of Gauss's in Which this estimate
occurs is entitled "Einige Bemerkungenzu Vega's Thesaurus LogarithmO^
rum," and appeared in the ' Astronomische Kachrichten,' 'No. 756, for May 2,
1851 (reprinted ' Werke,' t. iii. pp. 257-264), It contains an examination
of the relative numbers and magnitudes of the last-figure errors that occur
in the. sine, cosine, and tangent columns. It is easily shown that the tan-
gents were formed by mere subtraction from the sine and cosine columns j
but Gauss was unable to explain the fact that the cosines were more accu-
rate than the sines, which appeared as one of the results of the examination.
This question is further discussed in the ' Monthly Notices of the' Roy. Ast.
SoCi ' for May 1873 ; and it is there shown by the reporter that this result is
a direct consequence of the formula by means of which Vlaoq calculated the
table. So long as all these errors remain uncorrected, the logarithmic trigo-
nometrical canon cannot be considered to be in a satisfactory state, as it is
certainly desirable that a reliable ten-place table should exist.

We believe no perfect list of errors in Vega has been given : a number of
errors in T. I. are given by Lefort (' Annales de I'Observatoire de Paris,'
t. iv.) ; but this list could not, from the manner' in which it was formed, in-
clude any errors that did not also occur in Viacq.

A long list of errors in the trigonometrical tables of Vega is given by
Gronau, ' Tafeln fiir die hypeJbolisohen Sectoren' &c. Dantzig, 1862, p. vi.

Copies of Vega are still procurable (but with difficulty and delay) from
Germany, through a foreign bookseller, for about £1 10s. or £1 15s.

Vega (Manuale), 1800. T. I, Seven-figure logarithms to 1000, and
from 10,000 to 101,000, with proportional parts. The change in the line
is denoted by an asterisk prefixed to the fourth figure of all the logarithms
affected. A few constants are given on p. 188.

T. II. Log sines, tangents, and arcs for the first minute to every tenth of
a second. Although there is a triple heading, there is but a single column of
tabular results, as lor such small angles the sines, tangents, and arcs are equal
to one another,

140 BEPOKT— 1873.

Log sines, cosines, tangents, and cotangents, from 0° to 6° 3' at intervals
of 10"^, and thence to 45° at intervals of 1', to 7 places, with differences for
1" throughout.

An Appendix contains some spherical trigonometry. One page (p. 297)
contains longitudes of arcs, viz. circular measure of 1°, 2°, . . . . 90°, ana
by intervals of 10° to 180° ; also of 360°, of 1', 2', . . . . 60', and of 1", 2", ....
60", to 8 places. At the end some errata are given, and also some in Caliei
and other works.

The description of this work, according to order of date, should follow the
next ; but as it is referred to in the latter it is convenient to place it first.

Vega (Tabulae), 1797. Vol. i. — T. I. is identical, page for page, with
T. I. of Vega's 'Manuale ' just described, and was most likely printed from
the same type. The constants &c. on p. 188 are also identical,

T. n. is also identical with T. II. of the ' Manuale,' only with the addition
of 40 more pages, containing log sines and, tangents from 0° for every
second to 1° 30' 0", to 7 places, without differences. Thus the ' Tabulae ' and
the ' Manuale ' agree to p. 193 ; then the 40 pp. are inserted in the ' Tabulae,'
and pp. 233-330 of the ' Tabulae' are identical with pp. 193-290 of the
' Manuale,' the coincident portions of tfie two works being doubtless printed
from the same type.

T. III. Natural sines and tangents to every minute of the quadrant, to
7 places, with differences for one second throughout.

The Appendix contains a table of circular arcs, viz. the circular measure
of 1°, 2°, 3°, . . . . 360°, of 1', 2', . . . . 60', and of 1", 2" 60" (with the cor-
responding number of seconds in tliese angles), to 8 places, and small tables
for the conversion of arc into time, and hours &c. into decimals of a day. On
pp. 407-409 are given one or two constants connected with the calcula-
tion of TT, the values of a few radicals, and the expression for the sine of
every third degree in radicals. Some errata are given at the end of the
introduction.

Vol. ii. — ^T. I. Table of all the simple divisors of numbers below 102,000
(2, 3, and 5 excluded) ; a, b, c, d are printed for 11, 13, 17, 19, to save room.
This is followed by primes from 102,000 to 400,000. Ckeknac (§ 3, art. 8)
found 39 errors in this table : see his preface.

T. II. Hyperbolic logarithms of numbers to 1000, and of primes from 1000
to 10,000, to 8 places. This table is followed by the first 45, 36, and 27
powers of 2, 3, and 5 respectively.

T. III. gives c-^ and Briggian log ^ (the former to 7 figures, the latter to 7
places), from a;=0-00 to .r= 10-00 at intervals of -01.

T. IV. The first nine powers of numbers from 1 to 100, squares from 1
to 1000, cubes from 1 to 1000, and square and cube roots of numbers from
1 to 100, io 7 places.

T. V. Logistic logarithms, viz. log 3600— log (number of seconds in argu-
ment), for every second to 1° (=3600"), to 4 places.

[T. VI.] The first six binomial-theorem coefficients, viz. x, -^ — -^ . , ,;

'-^^j — ._,'"" „ , from x=-Ql to a;=l-00 at intervals of "01, to 7 places.

113 1 1 '^

This is followed by a page of tables, giving ^-j, g" jyg 2~s> 2 4 5 '

1 3
.... 2~A *"■> ^ 1*^ places, with their logarithms to 7 places.

ON MATHEMATICAL TABLES. 141

The rest of the book is devoted to astronomical tables and forranloe, except
two remarkable tables at the end (pp. 364-371). The first of these [T. VIl.]
is most simply described by stating that it gives the number of shot in a py-
ramidal pile on a square base, the number n of shot in the side of the base
being the argument ; the table extends from 71= 2 to ii = 40. There is also
given the number of shot in a pyramidal pile on a rectangular base, the ar-
guments being n the number of shot in the breadth of the base, and m the
number of shot in the top row (so that m+n — 1 is the number in the length
of the base). The ranges are, for n, 2 to 40, and form, 2 to 44, the table being
«f double entry.

[T. VIII.] gives the number of shot in a pyramidal pile on a triangular
base, the number of shot in a side of the base being the argument, which
extends from 2 to 40. The other portion of the table is headed " Tabula
pro acervis globorum oblongis, ab utraque extremitate ad pyramides quadri-
lateras appositis;" and the explanation is as follows: — Suppose we have
two pyramidal piles of shot on square bases (n shot on each side) placed
facing one another, at a distance equal to the sum of the diameters of m shot
ia,part ; and suppose it is required to fill this interval up, so as to make a pyra-
midal pile on a rectangular base, then this table gives the number for n (latus)
to M=40, and for m (longitmio baseos) to m=4:4:, the table being of double
entry.

Some errata are given after the introduction.

"We have seen the third edition (Leipzig, 1812) ; and though we have not
compared it side by side with the second (here described), we feel no doubt
the contents are identical ; at all events the number of pages in each volume
8 the same, and the preface is dated 1797 in both.

Vlacq (Arithmetica Logarithmica), Gouda, 1628, and London, 1631,
1_T. I.] Ten-figure logarithms of numbers from 1 to 100,000, with difier-
ences. This table occupies 667 pages.

[T. II.] Log sines, tangents, and seeants for every minute of the quadrant^
to 10 places, with interscript differences ; semiquadrantally arranged. This
table occupies 90 pp.

In the English copies, by George Miller, there is an English introduction
of 54 pp.j and then follows a table of latitudes (8 pp.). The original edition
of 1628has 79 pp. of introduction ; and a list of errata is given, which does
not occur in Miller's copies (but see ' Monthly Notices of the Eoy. Ast. Soc.'-
t. xxxiii. pp. 452, 456, May, 1873).

There were also copies with a Erenoh titlepage ; and in these there is an
Introduction in the same language of 84 pp. We suspect that a Dutch edition
was contemplated, but that the copies of the table intended for this purpose
afterwards formed Miller's English edition : no Dutch edition is known to
exist (see Phil. Mag., May 1873). The titles of the three editions are given
in 'full in § 5 ; in all, the tabular portion is from the same type. The bibli-
ography of this work forms an essential part of the history of logarithms ; and
a good many of the references occurring in the introductory remarks to § 3,
art. 13, have reference to it.

The table of logarithms of numbers contains about 300 errors, exclusive
of those affecting the last figure by a unit ; but a good many of these have
reference to the portion below 10,000, -which need never be used. This is
still the most convenient ten-figure table there is (VEeA,fol. 1794, is the only
other) ; but before use the known errata should be corrected. Eeferences to
all the places where the requisite errata-lists are to be found are given in the
' Monthly JiTotices of the Eoy. Ast. Soc' for May and June, 1872. We intend, ,

112 REPORT— 1873.

however, in the next Beport to give a complete list of errors in the portion
of the table from 10,000 to 100,000.

"We succeeded in obtaining a copy of this -work after some difficulty ; Mr.
Merrifield informs us that copies have always been procurable from abron(i
for about £2.

Vlaeq (Trigonometria Artificialis), 1633. [T. I.] Log sines and tan-
gents to every ten seconds of the quadrant, to 10 places, with characteristics
and differences (not interscript) ; seroiquadrantaUy arranged. The table
occupies 270 pp.

[T. II.] Ten-figure logarithms of numbers to 20,000, with differences,
printed from the same type as that used in the * Arithmetica '(1628 and 1631)
(except the last 500). A list of errata is given on the last page, The trigo*
nometry &c. at the beginning occupies 52 pp. See § 3, art. 15 (introductory
remarks), and also Vesa (fol.), 1794.

Vlacq, 1681. This is one of the numerous small editions caUed after
Vlacq, on the Gellibrand model. The contents, shape of type, &b, are exactly
the same as in Hentscicen- (Vlaoq), 1757, § 4, except that in the latter the
"whites" are rather wider. The printed portion of the page of tables is
3|- in. by 5| in. There are 48 pp. of trigonometry &c. in Latin. No name
except Vlacq's appears in connexion with the work.

[T. I.] Natural sines, tangents, and secants, and log sines and tangents
for every minute, to 7 places.

[T. JI.] Logarithms of numbers from 1 to 10,000, arranged conseoiitively
in columns, to 7 places ; no differences.

In one of the copies we have seen there are several errors corrected in
manuscript. This edition must be rather common in England, as we hi»vo
seen several copies,

Wackerbartb} 1867. T. I. Five-figure logarithms (arranged as in,
seven-figure tables) to 100, and from 1000 to 10,000, with proportional
parts .to tenths (i. e. multiples of the differences). The degrees, minutes, &c.
corresponding to eight numbers on the page are given at the bottom of each,
At the end of this table there are added seven-figure logarithms of numbers
from 10 to 100, and also from 10,000 to 11,000, the latter with proportional
parts to tenths.

T. IL Log (1.2.3. .,.«) for fl;=l,2,,, ..100; log (1.3.5. ,..») for
a?=l, 3, 5 65 ; log (2 , 4 . 6 . . . , a;) for .i;=2, 4, 6 , . , . 66 : all to 5 places.

T. III. Log sines and tangents for every second from 0' to 10' ; log sines and
tangents for every ten seconds from 0° to 5° ; log sines and tangents for every
minute of the quadrant : all to 5 places. Differences are added throughout,"
and also proportional parts to tenths (i. e. multiples of the differences) for every
second to 5°, and for every 10 seconds in the other portion of the table.

T. IV. Circular measure of 1°,2°,. . , .180°, of l',2',. . . .60', and of 1",
2", .... 60", to 5 places. Some constants, such as the unit arc, its logarithm
&o., are added.

T, Y, Hyperbolic logarithms of numbers from 1 to 1010, to 5 places, with
proportional parts to tenths, arranged as in seven-figure tables of Briggian
logarithms ; followed by the first hundred multiples of the modulus and its
reciprocal, to 5 places, A few constants, tt, e, &c., are given, to 30 places.

T. VI. Squares of numbers from 1 to 1000.

T, VII. Square roots (to 7 places) of numbers from 1 to 1000.

T, VIII. Natural sines, cosines, tangents, and cotangents for every 10'
to 5°, thence for every 20' to 15°, and thence to 45° at intervals of 30', to 3,
places.

ON MATHEMATICAL TABLES. 143

T. IX. Reciprocals (to 7 places) of numbers from 1 to 1010.

T. XVII. List of primes to 1063.

T. XXI. gives some constants.

The other tables are chemical &o,

This is one of the most complete fiye-figuro tables wo have seen. The
change in the leading figures, where it occurs in the middle of a Uno, is
throughout denoted by an asterisk prefixed to the third figure of all the
logarithms afiected. It may be remarked that though the introduction &c. is
in Swedish, the headings of the tables are in Latin.

A list of four errata in the tables is given- by Prof, Wackerbarth himself
in the ' Monthly Notices of the Royal iitronomical Society,' t. xxxi. No. 9
(Supplementary Nuiiiber, 1871).

Wallace, 1815. [T. I.] Six-figure logarithms to 100, and from 1000 to
10,000, with differences,

[T. IL] Log sines, tangents, and secants to every minute of the quadrant,
to 6 places, witlf differences.

[T. IIIiJ Natural sines to evej-y minute of the quadrant, to 5 places. This
is followed by a traverse table.

The tables are preceded by 148 pp. of trigonometry &c,

Wamstorff's Schumacher, 1845, Out of 221 pages, only 21
(pp. 116-120 and 206-221) come within the scope of this Report,

[T. I.] For the conversion of arc into time, and vice versd.

[T. II.] The circular measure of 3°, 2=,. , , .90°, 95°. . . .120°, 180°. . . .
360°, of 1', 2' 60', and of 1", 2", .... 60", to 7 places.

[T. III.] Four-figure logarithms to 1009.

[T. IV.] Log sines, cosines, tangents, and cotangents at intervals of 4'
to 10°, and thence to 45° at intervals of 10', to 4 places.

[T. v.] Gaussian logarithms ; B and C are given for argument A from A =
•00 to 1-80 at intervals of -01, and thence to 4-0 at intervals of -1, to 4 places,
with differences.

The other tables are astronomical.

WilUch, 1853, T. XX. Seven-figure logarithms to 1200, followed by a
few constants, &e,

T. XXI, Squares, cubes, square and cube roots (to 7 places), and reci-
procals (to 9 places) of numbers to 343, followed by some constants.

T. A. Hyperbolic logarithms of numbers from 1 to 1200, to 7 places.

T. B, Natural and log sines, tangents, secants, and versed sines, for every
half degree, to 7 places,

T. C, Circumferences and areas of circles for a given diameter, viz. nd

(to 5 places) and V" (to 2 places) for ^=5=1, 2,. . , .9, and from (Z=l to

100 at intervals of -25.

T, D. Circular measure of 1°, 2°, 180°, to 7 places.

The other tables iu the work are of a very varied character.

We have also seen the second edition (1852), which does not contain the
tables A to D ; and we nave seen a review of the seventh edition, edited by
M. Marriott, 1871,

§ 5, lAst ofiuorks containing Tables that are described in this Report, with refer-
ences to the section and article in which the description of their contents is
to he. found.
[Those works to which an asterisk is prefixed have not come under the'

inspection of the reporter ; and the desoription of their contents is therefore

144 REPORT — 1873.

derived from some secondhand source. The author's name is enclosed within
square brackets when it does not occur on the titlepage of the work. For other
explanations see § 2, arts. 4-14, and § 6 (Postscript), arts. 2-4, 8, 10-12.]

AcAD^MiE EoTAiE . . . DE Prtjsse, PublLe sous la direction de 1'. Eecueil
de Tables Astronomiques. Berlin, 1776. 3 vols. 8vo. § 4. .

Adams, John. The Mathematician's Companion, or a Table of Logarithms
from 1 to 10,'860 . . . London, 1796. Svo. § 4.

AiRT, (J. B., Com])uted under the direction of; Appendix to the Greenwich
Observations, 1837. London, 1838. 4to. § 3, art. 15.

Alsteditts, J. H. Scientiarum omnium encyclopaediie tomus primus . . ,
Lugduni, 1649 (2 vols. fol.). § 3, art. 4.

Andrew, Jambs. Astronomical and Nautical Tables, with Precepts , . ,
London, 1805. 8vo (pp. 263). § 4.

Anontmotjs. Multiplieationstabelle, enthaltend die Producte allor ganzen'
Factoren von 1 bis 1000, mit 1 bis 100, Kopenhagen, 1793. 4to (pp. 247 ;
and introduction, pp. 8). § 3, art. 1.

ANomrirotrs. Tables de Multiplication . . . Paris, 1812. § 3, art. 1.

Anontmotts. TafellogistischerLogarithmen. Zugabe zu den Vega-Hiils-
se'schen und anderen Logarithmen-Tafeln. Aus CaUet's " Tables de Loga-
rithmes." Niimberg. Verlag von Eiegel & Wiessner. 1843 (table, 7 pp.).
§ 3, art. 18.

Anontmotjs (1844). See Sheepshah-ks.

AjfONYMOTTS. Logarithmen. Antilogarithmen. Berlin. [On a card, 1860 ?]
§4. , .- ,

AtTxixiAKY Tables. See [Scettmacheu.]

Babbaoe, CsaKles. Table of the Logarithms of the Natural Numbers from
1 to 108000. . . Stereotyped. Fourth impression. London, 1841 (202 pp. and
explanations &c. xx). § 3, art. 13.

[The 1838 edition (or rather tirage) has the following notice of errata
contained in it, on the back of the titlepage : " In the logarithms of 10354,
60676 to 9, 70634 to 9, and 106611 to 9, the fourth figures ought to be
small instead of large. In the list of constants the last figure of the value
of e should be 8 instead of 9." The tables were stereotyped from their first
publication in 1827. Mr. W. Barrett Davis has called our attention to the
number of last- figure unit errors in the portion of the table beyond 100,000 ;
thus on p. 192 there are no less than fifteen such errors which are corrected
in more recent works, such as Scheon and Kohlee. This portion of the
table Babbage copied from Callet.]

Babbaoe CATALootJE, Mathematical and Scientific Library of the late
Charies Babbage of No. 1 Dorset Street, Manchester Square. To be sold by
Private Contract. . . . Printed by C. F. Hodgson and Son, Gough Square,
Fleet Street [London], 1872. [The catalogiie was drawn up by Mr. llobert
Tucker, M.A., Honorary Secretary of the LbiidonMathematical Society ; and
the library was purchased by Lord Lindsay.]

- BaoIt, V. Nouvelles Tables Astronomiques et Hydrographiques . , . .
Edition stereotype. . . Paris, Firmin Didot, 1829. Small 4to. § 4.

Baklow, Peter. New Mathematical Tables containing the factors, squares,
cubes, square roots, cube roots, reciprocals, and hyperbolic logarithms of all
numbers from 1 to 10,000, '. . . " London, 1814. Svo (pp. 336, and intro-
duction Ixi). § 4.

Baklow's Tables of Squares, Cubes, Square roots, Cube roots. Reciprocals
of all integer numbers up to 10,000. ■ Stereotype edition, examined and cor-
rected. (Under the Superintendence of the Society for the Diffusion of Usefu '

ON MATHEMATICAL TABLES, 143

Knowledge.) London, 1851, from the stereotyped plates of 1840. 8vo ("pp.
200). § 3, arts. 4 and 7.

Bates, Dayid. Logarithmic Tables, containing the logarithms of all num-
bers froni 1 to 10 000, together with . . . Dublin, 1781. (63 pp. of tables,
introduction ccxi pp., and appendix 60 pp.) § 4.

Bbardmoke, Nathaniel. Manual of Hydrology : containing . . . London,
1862. Bvo (pp. 384). § 4.

Beknotoli, John. A Sexcentenary Table . . . Published by order of the
Commissioners of Longitude. London, 1779. 4to (pp. 165; and intro-
duction, -viii). § 3, art. 9.

Beethottd, F. Les Longitudes par la mesure du temps . . . Paris, 1775.
Small 4to (34 pp. of tables). § 3, art. 15.
Bbssel. See [Schtjmaohee.]

Beverley, Thomas. The Mariner's Latitude and Longitude Eeady-oom-
puter . . . Cirencester (no date; but Appendix dated 1833). 4to(pp. 290). §4.
Blanchaed. See Gaedinbe (Avignon edition, 1770).
, BoNNYCAsiLE, John. An Introduction to Mensuration .... The fifteenth
edition . . . London, 1831. Small 8vo. § 3, art. 22.

BoEDA, Ch. Tables trigonometriques decimales ou Tables des logarithmes
. . . revues, augmentees et publiees, par J. B. J. Delambee. Paris, An ix.
[1800 or 1801]. SmaU 4to. § 4.

BowDiicH, N. The improved Practical Navigator ; ... to which is added
a number of new Tables .... Revised, recalculated and newly arranged by
Thomas Kiebt. London, 1802. 8vo. § 4.

Bremikee, C. Tafel der Proportionaltheile zum Gebrauche bei Ipgarith-
mischen Bechnungen mit besonderer Beriicksichtigung der Logarithmentafeln
von Callet und Yega. . . BerUn, 1843. 8vo (pp. 127). § 3, art. 2. _

Bremikee, C. Logarithmorum VI decimalium nova tabula Berolinensis . . .
Berolini, 1852. 8vo. § 4.

Beemieer's Veoa. See Yesa (1857).
Bremikee. See Ceelle (1864).

Bretsohneidee, C. a. Produktentafel enthaltend die 2, 3 .... 9 fachen
aller Zahlen von 1 bis 100 000. Hamburg und Gotha, 1841. 8vo (pp. 110),
§ 3, art. 1.

BuiGSE, H. Tables des Logarithmes . , , 1626. See under de Decker,
1626, § 4.

[Beigos, Henet.] Logarithmorum Chilias Prima. [London, 1617.] SmaU
8vo (pp. 16). § 3, art. 13.

BuiGGS, Henet. Arithmetica logarithmica sive logarithmorum chiliades
triginta, pro numeris natural! serie crescentibus ab unitate ad 20,000 : et a
90,000 ad 100,000. Quorum ope multa perficiuntur Arithmetica problemata
et Geometrica. Hos numeros primus invenit clarissimus vir lohannes Nepe-
rus Baro Merchistonij ; eos autem ex eiusdein sententia mutavit, eorumque
ortum et usum illustravit HenricusBriggius, in celeberrima AcademiaOxoniensi
Geometriie professor Savilianus. Deus nobis usuram vitse dedit et ingenii,
tanquapi, pecuniae, nulla prsestituta die. [Eoyal arms, I. R.] Londini, Ex-
cndebat (Gulielmus lones, 1624. folio (preface &o. 6pp., trigonometry 88 pp. ;
tables unpaged). § 3, art, 13, • ■ _

(Some copies of this work were also published in 1631, with the same title-
page as Ylacq's LogarithmicaU Arithmetike. See § 3, art. 13.)

Beiggs, Henet. Trigonometria Britannica : sive de doctrinatriangulorum
libri duo. Quorum prior continet Constructionem Canonis Sinuum Tangen-
tium &, Secantium, una cum Logarithmis Sinuum & Tangentium ad Gradus
1873. -^

146 KEPOBT — 1873.

& Graduum Centesimas & ad Minuta & Secunda Centesimis respondentia : A
Clarissimo Doctissimo Integerrimoque Viro Domino Henrico Briggio Geome-
trise in Celeberrima Academia Oxoniensi Professore Saviliano Dignissimo,
paulo ante inopinatam Ipsius e terris emigrationem compositus. ■ Posterior
vero usum sive Applicationem Ganonis in Eesolutione Triangulorum tam
Planorum quam SphsericoriTm e Geometricis fundamentis petitft, calculo faciJ-
limo, eximiisque compendiis exMbet : Ab Henrico Gellibrand Astronomiae in
Collegio Greshamensi apud Londinenees Professore construe tus. [Then follow
a quotation of three lines from Vieta and a diagram showing the trigonome-
trical functions.] Goudae, Excudebat Petrus Eammasenius. m.do.xxxiii.
Cum PrivUegio. folio. (Dedication to the Electors to the Savilian Chairs,
Gellibrand's preface, and 110 pp. of trigonometry &o., followed by one page
containing errata to the page_ signature /. 3 of the tables; the tables are
unpaged.) § 3, art. 15.

BBifiGS. See Sherwin.

Browk. See Wamaoe.

Bhowne, Eobekt; a new improvement of the Theory of the Moon ....
London, 1731. SmaU 4to (pp. 1 4). § 3, art. 25.

Beuhns, Dr. A new Manual of Logarithms to seven places of Decimals ....
Stereotype edition. Bernhard Tauchnitz. Leipzig, 1870. 8vo (pp. 610, and
introduction xxiii). § 4.

Beuwo, Faa db. Txait^ ^l^mentaire du Calcul des Erreurs avec des Tables
stereotypies , . . Paris, 1869. 8vo (41 pp. of tables). § 3, art. 4.

Bubokhaeut, J. Ch. Tables des Diviseurs pour tous les nombres du dcuxi^me
million . . . Paris, 1814. 4to (pp. 112 and viii). § 3, art. 8.

Bttrckhaedt, J. Ch. Table des Diviseurs pour tous les nombres du troisi^me
million . ; , " Paris, 1816. 4to (pp. 112). § 3, art. 8.

BtrRCKHAEDT, J. Ch. Table des Diviseurs pour tous les nombres du premier
million . . . Paris, 1817. 4to (pp. 114, and preface &o. 4 pp.). § 3, art. 8.

*BtEeEE, J. A. P. Tafel zur Erleichterung in Eechnungen &c. 1817. See
under CEunfEESCHWEE, 1825, § 3, art. 3.

Btene, Olivee. Practical, short, and direct Method of calculating the
Logarithm of any given Number, and the Number corresponding to any given
Logarithm, discovered by Oliver Byrne . . . London, 1849. 8vo (pp. 82, and
introduction xxiii). § 4.

BiEiTB, Olivee, Tables of Dual Logarithms, Dual Numbers, and corre-
sponding Natural Numbers ; with proportional parts of differences for single
digits and eight places of decimals . . . London, 1867. Large 8vo (pp. 202,
and introduction pp. 40). § 3, art. 23.

Btkne, Olivee. Other works. See § 3, art. 23.

Callet, FEAirgois. Tables portatives de Logarithmes, contenant ....
Edition stereotype, gravde, fondue et imprim^e par Firmin Didot, Paris ;
Firmin Didot, 1795 (Tirage, 1853). 8vo (pp. 680, and introduction pp. 118).
§4.

Callet, F. Table of the logarithms of sines and tangents .... Paris,
1795 (Tirage, 1827). Stereotyped and printed by Firmin Didot .... 8vo.
§ 3, art. 15.

Callet (1843). See Akontmotts,

Cexuteeschweb, J. J. Neu erfundene Multiplikations- und Quadrat-Tafeln
. . . mit einer Vorrede von ... J. P. Gxuson und L. Ideler. Berlin, 1825.
8vo (45 pp. of tables, and introduction Iv). § 3, art. 3.

Cheenac, Ladislaus, Cribrum Arithmetieum ; sive tabula continens nu-
jneros primos . , . Daventrise, 1811.- 4to (pp. 1020), §3, art. 8,

ON MATHEMATICAL TABLES. 147

*Clouth, p. M. Tables pour le Calcul des Coordonn^es gomom^triques.
Mayen (chez I'auteur). 8vo. § 3, art. 10.

Coleman, Geoeob. Lunar and Nautical Tables .... Stereotype edition.
London, 1846. 8va (317 pp. of tables). § 4.

Ceelle, a. L. Erleicbterungs-Tafel fiir jeden, der zu rechnen hat ; enthal»
tend die 2, 3, 4, 5, 6, 7, 8, und 9 facHcn aller Zahlen von 1 bis 10 MiUionen
... Berlin, 1836. (pp. 1000 and explanation xti.) § 3, art. 1.

Ceelib, a. L. Eechentafeln welche alles MultipUt;iren und Dividiren mit
Zahlen unter Tausend ganz ersparen . , . Zweite Stereotypy Ausgabe . . . von
Dr. C. Beemiker. Berlin : Georg Eeimer, 1864. Folio (pp. 450). [There is
also a French titlepage.] Also edition of 1820, in two' vols. 8vo. § 3, art. 1.

Ceoswell, WiLzi&.M. Tables for readily oomputing the Longitude ....
Boston, 1791. 8vo. § 4. .

Dase, Zachaeias. Tafel der natiu-Hchen Logarithmen der Zahlen. In
der Forin uiid Ausdehming wie die der gewohnlichen oder Brigg'schen
Logarithmen.. . Wien, 1850. 4to (pjp. 195)< • § 3, art. 16.

Dasb, Zachaeias. Factoren Tafeln fiir alle Zahlen dor Siebenten Million .
... Hamburg, 1862. 4to (pp. 112). § 3, art. 8.

Dasb, Zaghamas. Factoren Tafeln fiir alle Zahlen der Achten Million. . . .
Hamburg, 1863. 4to (pp. 112). § 3, art. 8.

Dase, Zachaeias.. . Factoren-tiafeln fiir Zahlen der Neunten Million, i.
erganzt von Dr. H. Eosenbeeg. Hamburg, lg65j 4to (ijp. 110). § 3, art. 8.

Dbchales (Cursus Mathematicus)." §2, art. 3.
, De Deckee. Nieuwe Telkdnst, inhbiidende de Logarithm! vopr de Ghetallen
beginnende van 1 tot 10000. . . Door Ezechiel db Decker, Eekenmeester,
ende Lantmeter residerente ter Goude . -. . Ter Goude. By Pieter Eammaseyn
. . . 1626. 8vo (260 pp. of tables, and introduction pp. 50+, (copy imper-
fect)). [De Haan gives 61 as the number of ppi in- the introduction, ' Phil.
Mag.' May, 1873]. §4. - •■

Degen, C. F. Tabularum ad faciUorem et breviorem Probabilitatis com-
putationem ufilium Enneas .... Havni», 1824. 8vo (pp. 44, and intro-
duction xxii). § 4.

De Haan (lets over Logarithmentafels). § 3, art. 13 (p. 55).

De Joncottkt. See Joncoitet.

Db la Lande. See Laiandb.

Delaubee. See Boeda.

De Meitdoza. See Eios.

Db Monti'eeeiek. See Montpeeeiee.

[De Moegan, a.]. Tables of Logarithms (Under the superintendence of
the Society for the Difiusion of Useful Knowledge). London, 1854. From
the stereotyped plates of 1839. Small 8vo (pp. 215). § 4.

De Moesan, a. Encyclopaedia Metropolitana. Pure Sciences, vol. ii.
{Theory of Probabilities}. London, 1843. § 3, art. 25.

Db Moegan (Article on tables in the Penny and English Cyclopasdias and
' Arithmetical Books '). § 2, art. 3.

Db MoRGAif. See Scheon (1865).
. Db Peassb. Tables logarithmiques, pour les nombres, les sinuS et leS
tangentes, disposees dans un nouvel ordre . , . Accompagn^e de notes et d'ntt
avertissement par M. Halma. Paris, 1814. 12mo (pp. 80). §4.

Dessiotj. See J. H. Mooeb.
- Dilling, J. M. Probeschrift eines leichtfasslichen logarithmischen Sys-
tems. .,. .fur Burger und Landsohulen . . . . Leipzig, 1826. 12mo (pp. 53)v
§ 3, art. 1.

1.2

]4g REPORT — 1873.

Donsos, Jamt!3. The Antilogarithmic Canon .. . London, 1742. folio. §3,

DoDSON, Jame3. The Calculator : being correct and necessary tables for
computation. Adapted to Science, Business, and Pleasure London, 1747.

Large 8vo (pp. 174). § 4, .,.,„, ^ , ,.■

DoMKE, I". Nautische astronomische und loganthmische lateln...lur
die Koniglich Preussischen Navigations-Schulen . . . Berlin, 1852. 8vo
(353 pp. of tables). §4.

DoNN, Benjamin. Mathematical Tables, or Tables of Logarithms . . . Ihird
edition, with largo additions. London, 1789. 8vo (pp. 351). § 4.

DoTioLAs, George. Mathematical Tables, containing the Logarithms of

Numbers ; Tables of Sines, Tangents, and Secants and Supplementary

Tables. Edinburgh, 1809. 8vo (pp. 166). § 4.

DoirwES. See under Bowditch, § 4.

DirooM, P. Cours d'Observations nautiques, contenant . . .suivi d'une col-
lection desmeiUeu res Tables. .. Bordeaux, 1820. 8vo (296 pp. .of tables). §4.

Dumas. See Garmneb (Avignon edition, 1770).

Dunn, Samuel. Tables of correct and concise logarithms for numbers,
sines, tangents, secants... London, 1784. 8vo (pp. 144). §4.

Dupuis, J. Tables de Logarithmes h, sept decimales d'apr^s Bremiker,
Callet, Vega, etc. par J. Dupuis. Edition st^rdotype .... troisieme tirage.
Paris, 1868. . 8vo (pp. 578). § 4.

Dupuis. See under Callet, 1853. § 4. _

[Enckb J. P.] Logarithmen von vier Decimal-Stellen. Berlin, 1828.
Small 8vo (pp.'22). §4.

Ebsch (Litteratur der Mathematik), § 2, art. 3,

EvERBiT, J. D. Universal Proportional Table .... William Mackenzie.
London [no date, 1866]. § 4.

Farley, Eichabd. Tables of six-figure logarithms . . . Stereotjrped edition.
London, 1840. 8vo. § 4,

[Farley, E.] Natural versed sines from 0° to 125°, and Logarithmic
versed sines from 0° to 135°, or O"" to 9'', used in computing Lunar Distances
for the Nautical Almanac. London : Eyre and Spottiswoode, 1856. folio
(pp.90). §4. _ ,

Faulhatjee, Johann. Ingenieurs-Schul, Erster Theyl : Darinnen durch
den Canonem Logarithmicum alle Planische Triangel zur fortification . . . . zu
solvireu . . . Auss Adriano Vlacq, Henrico Briggio, Nopero, Pitisco, Bemeck-
hcro . . . gezogen . . . Gedruckt zu Franckfurt am Mayn . , . 1630. Small 8vo (pp.
170) (with an Appendix of 14 pp.). Followed by an engraved titlepage. § 4.

[Faulhabbr, J.] Zehentausend Logarithmi der Absolut oder ledigen Zahlen ,
von 1. bias auff 10000. nach Herrn Johannis Neperi Baronis Merchistenii
Arth und Inuention, welche Heinricus Briggius illustriert, und Adrianus
Vlacq augiert, gerichtet. Gedruckt zu Augspurg, durch Andream Aperger,
auff unser heben Frawen Thor. Anno m.dc.xxxi. Small 8vo (pp. 104). § 4.

[Faulhabeb, J.]. Canon Triangulorum logarithmicus, das ist : Kiinstlicho
Logarithmische Tafeln der Sinuum, Tangentium und Secantium, nach Adrian!
Vlacqs Calculation Eechnung und Manier gestelt. Gedruckt zu Augspurg,
durch Andream Aperger, auff unser lieben Frawen Thor, Anno m.dc.xxxi.
Small 8vo (pp. 190). § 4,

Felkel, AiraoN. Tafel aUer Einfachen Factoren der durch 2, 3, 5 nicht
theilbaren Zahlen von 1 bis 10 000 000. I. Theil. Enthaltend die Factoren
von 1 bis 144000. . . , Wien, mit von Eheleuschen Schriften gedruckt, 1776.
targe folio (pp. 26, and preface, &c. 4 pp.). § 3, art. 8.

ON MATHEMATICAL TABLES. 149

Telkel. See Lambekt.

FnipowsKi, HEBscnELL E. A tal)le of Anti-logarithms, containing to seven
places of decimals, natural numbers, answering to alllogarithms from -00001

to -99999, and an improved table of Gauss's logarithms London, 1849,

8vo (pp. 220, and introduction xvi). § 4.

PiLipowsKi, H. The wonderful canon of logarithms ... by John Napier
.... retranslated from the Latin text, and enlarged, with a table of hyper-
bolic logarithms to all numbers from 1 to 1201. By Herschell Filipowski
.... Edinburgh, 1857. 16mo. § 3, art. 16.

PiNCK. Thomse Einkii Flenspurgensis Geometriaj rotundi Libri xiv. ad
Eridericum Secundum, Serenissimum Dani8e,.& Norvegise regem &c. Cum
Gratia & Privileg. Cses. Majest. Basilese per Sebastianum Henriepetri [15831.
4to. §3, art. 10.

Eiscebk's Vega. See Yeoa.

Eeench Manttsceipt Tables. See Tables dv Cadastre.

Galbbaith, D. The Piece-Goods Calculator, consisting of a series of tables
. . . Glasgow, 1838. 8vo (pp. 53). § 3, art. 25.

Galbbaith, J. A., and S. HATj&Hioif. Manual of Mathematical tables . . .
London, 1860. SmaU 8vo (pp. 252). § 4.

Galbbaith, William. Mathematical and Astronomical Tables . . . Edin-
burgh, 1827. 8vo (112 pp. of tables). § 4.

Gaedistee, William. Tables of Logarithms for aU numbers from 1 to
102100, and for the Sines and Tangents . . . London, 1742. 4to. § 4.

Gakdhjee, W. Tables de Logarithmes, contenant les Logarithmes des
nombres . . . des sinus & des tangentes . . . Nouvelle edition, Augmentee des
Logarithmes des sinus & tangentes pour chaque seconde des quatre premiers
degres. Avignon, 1770. 4to. (This reprint was edited by Pezenas, Dumas,
and Blanohakd.) § 4,

*GAEDiifEE. Paris edition, 1773. § 4.

Gaeeaed, William. Copious trigonometrical tables .... intended to com-
plete the requisite tables to the Nautical Almanack .... London, 1789;
8vo. § 4. .

Gauss, C. E. Tafel zur bequemern Berechnung des Logarithmen der
Summe oder Differenz zweyer Grossen, welche selbst nur durch ihre Loga-
rithmen gegeben sind. Zach's ' Monatliche Correspondenz,' t. xxvi. (pp. 498^
528). Gotha, 1812. § 3, art. 19.

Gauss. Carl Eriedrich Gauss Werke herausgegeben von der koniglichen

GeseUschaft der Wissenschaften zu Gottingen. Still in course of publication :
4to, t. i. (1863, and ' zweiter Abdruck,' 1870) ; t. ii. (1863) § 3, arts. 6 and
7 (introductory remarks) ; t. iii. (1866) § 3, art. 19 (introductory remarks) ;
and under De Peasse, HDlsse's Yesa, Pasquich, Yega (1794) in § 4 &c.
(t. iii. includes the reprints from the ' Astronomische Nachrichten ' and the
' Gdttingische gelehrte Anzeigen,' on logarithmic tables.) ;

Gellibband. See Beiggs (1633).

Gellibeand. See John Newton (1658).

Geenebth (Tract on the accuracy of logarithmic tables). Under Eheticus
(§ 3, art. 10), and § 3, art. 13 (introductory remarks, p. 55).

Glaishee, J. W. L. ' Monthly Notices of the Eoyal Astronomical Society : '
May, 1872 (On errors in Ylacq's (often called Briggs' or Neper's) table of
ten-figure logarithms of numbers) ; June, 1872 (Addition to a paper on errors
in Ylacq's ten-figure logarithms, published in the last Number of the ' Monthly
Notices ') ; March, 1873 (On the progress iio accuracy of logarithmic tables) ;
May, 1873 (On logarithmic taWes). ' Philosophical Magazine : ' October,

150 iiEPOKT— 1873.

1872 (Notice respecting some new facts in the early history of logarithmic
tables) ; December (Supplementary Number), 1872 (Supplementary remarks
on some early logarithaiio tables) ; May, 1873 (On early logarithmic tables
pnd their calculators). ' Messenger of Mathematics ' (new series) : (July,
1872 (Pineto's table of ten-figure logarithms of numbers) ; May, 1873 (Ee^
marks on logarithmic and factor tables, with special reference to Mr. Drach's
suggestions). § 3. art. 13 (introductory remarks ; Beiqgs, 1617 ; Pineio),
art. 15 (Gttntbr), art. 17 (Napieb, 1614), § 4, Borba and Delambee, db
Deckeb, HDmsb's Vega, Shoktkede, Vega, 1794, VLAca, 1633, &o.

[GoDWABD, William, Jun.] Interpolation tables used in the Nautical
Almanac Office. London: Eyre and Spottiswoode, 1857. 8vo (pp. 30),
§ 3, art. 21.

Gooi)WTif,'HEirRX. The first centenary of a series of concise and useful
tables of all the complete decimal quotients, which can .arise from dividing a
unit or any whole number less than each divisor, by all integers from 1 to
1024, [Loudon, Preface dated 1816], Small 4to (pp. 18 and introduction
xiv). § 3, art. 6.

GooDWTN, Henkt. The first centenary of a series of concise and usefiil
tables of all decimal quotients, which can arise from dividing a unit, or any
whole number less than each divisor, by all integers from 1 to 1024. To
which is now added a tabular series of complete decimal quotients, for all
the proper vulgar fractions, of which, when in their lowest terras neither the
numerator, nor the denominator is greater than 100 : with the equivalent
vulgar fractions prefixed. London, 1818, Small 4to (pp. 18 and 30, and
introductions xiv and vii). § 3, art. 6.

[GooDWTN, Henht.] a tabular series of decimal quotients for all the
proper vulgar fractions, of which, when in their lowest terms, neither the
numerator nor the denominator is greater than 1000, London, 1823. &va
(pp. 153 and introduction v). § 3, art. 6.

[GooDwnf, Henex.] A table of the circles arising from the division of a
■unit or any other, whole number by all the integers from 1 to 1024 ; being
all the pure decimal quotients that can arise from this source. London,
1823. 8vo (pp. 118 and introduction v). § 3, art. 6.

GoEDOir, James. Lunar and Time Tables .... for finding the Longitude
.... London, 1849. 8vo (92 pp. of tables). § 4.

Geaesse (Tresor de livres rares). § 2, art. 3,

Gkay, Petes. Tables and formiilse for the computation of life contin-
gencies . . . London, 1849. 8vo (68 pp. of tables). § 3, art. 19.

Ghat, Petbe. Addendum to tables and formulae for the computation of
life contingencies .... Second issue, comprising a large extension of the prin-
cipal table .... London, 1870, 8vo (26 pp. of tables) (noticed under the pre-
ceding work, § 3, art. 19). This title is copied from the wrapper of the
"Addendum," the titlepage of which is intended to apply to the whole work
when the " Addendum " is included, and runs, " Tables and formulae for the
computation of life contingencies .... Second issue, with an addendum, com-
prising a large extension of the principal table London, 1870."

Geat, Petbe. Tables for the formation of Logarithms and Anti-logarithms
to twelve places; with explanatory introduction. . . . London, 1865. 8vo
(55 pp. of introduction &c. and xi pp. of tables). § 3, art. 13.

Geegoet, OLDfiHirs. Tables for the use of nautical men, astronomers, and
others ; by Olinthtts Geegoet, "W. S. B. Woolhousb and James Hahit.
London, 1843, 8vo (pp. 168 and introduction xxiv), § 4,

Geegoey, OLijfiHus, See Hution (1858).

0:>J JJATHEMATICAL TABLES. 151

T '^™^^'*™K. Elomepta trigonometrica, id est sinus tang&ntes, secantes
la J/artibus Sums totius D.OOOOO. Christophori Grienbergeri E Societate lesu.
Keriim Mathematioarmn Opusculum Secundum. [Beyice— globe with IHS.]'
Komse, Per Hsered. Barthol. Zan. 1630. ■ Superiorum permiasu. 12nio (pre^"
tace and tables unpaged, trigoiiometry 88 pp., and 4 pp. of correqtiQns), § 3,

GrKiFMisr, James. A eomplete Epitome of Practical Navigation .... to
whioh IS added an extensive set of Kequisite • tables ... London, 1843.
8vo (325 pp. of tables). § 4.

GEUENBEEeEE, Grtjenpeesek, or Geiembergeb. See Gbienbeegeb.

— GKTrsoN-,_ J. P. Pinacotkeque, ou collection de Tables d'uue utilite gehe'rale
pour multiplier et divisor inventees par J. P. Gkuson. Avec une table de
tous les faeteurs simples de 1^10500. Berlin, 1798, 8vo (pp. 418 and
introdtiction xxiv). § 3, art. 1.

- Geuson, J. P. Grosses Einmaleins von Eins bis Hunderttausend. Erstes
Heft vons Eins bis Zehntausend . . . Berlin, 1799. Large folio (pp. 42).
§ 3, art. 1.

GETrsoif, J. P. Bequeme logarithmisobe, trigonometrischo und andere
niitzliche Tafeln zur Gebrauob auf Scbulen . . . J)ritte verbesserte Auflage.
Berlin, 1832. 8vo. § 4.

Gettsoh-. See Cbitineeschwee,

Gtottee, Edmttnj). Canon Triangulorum sive Tabute Sinuum et Tangen-
tium artificialium ad Eadium 10000,0000 & ad serupula prima quadrantis:
Per Edm. Gunibe, Professorem Astronomiie in CoUegio Greshap[ionsi. Londini,
excudebat Gulielmus Jones, mdcxx! Small 8vo(p. 94). § 3, art, 15.

Gtjntee, Edmund. The works of ; . . . with a canon of artificial sines and-
tangents . . . The fifth edition, diUgentlj corrected ... By 'Wimam Ley-
bourn, Philopaath. London, 1673. Small 4to. § 3, art, 15.
Hallet. See [SnEEwpr.]
Halma. See Db Peasse. '
Hann. See OLnfTHirs Geegoex (1843),

Hantschl, Joseph. Logarithmisch-trigonometrisches Handbucb .... Wien,
1827. Large 8vo. § 4.

Haktig, G. L. Kubik-Tabellen fiir geschnittene, beschlagene und runda
Holzer . . . und Potenz-Tabellen, zur Erleichterung der Zins-Berechnung . , ,
Dritte Auflage . . . Berlin und Stettin, 1829. 8vo, (pp. 488 and introduc-
tion xviii). § 4.

Hassiee, E. E.. Tabulae logarithmicse et trigonometriese, notis septem
decimalibus expressse, in forma minima . .,. Novi-Eboraci, 1830, 12mo
[stereotyped]. § 4.

Hasslee, F. E. Logarithmic and trigonometric tables, to seven places of.
decimals, in a pocket form . . , New York, 1830.- 12mo [stereotyped], § 4:
Hassxee, E. E. Tables logarithmiques et trigonometriques , a sept deci-
males, en petit format ., . NouveUe- York, 1830. 12mp [stereotyped]. §4.
Hasslee, E. E. Logarithmische und trigonometrischo Tafeln, zu sieben
Dezimal-Stellen ; in Taschen-Eormat . , . Neu-York, 1830. 12mo [stereo-
typed], §4.

Hassiee, E. E. Tablas logaritmicas y trigonometricas para las siete deci-
males, corregidas . . . Kueva-York, 1830. 12mo [stereotyped]. § 4.
Hattghtok-. See J. A. Gaibraith.

HEiiBRONiraE, C. Historia Matheseos Universse ... Lipsise, 1742, 4tp,
§ 3, art. 25 ; and see § 2, art. 3.

Heneioh", Denis. Traiote des logarithmes'. Par D. ^pieijrion, Profe?geur

152 REPORT — 1873.

es Mathematiqnes. [Typographical ornament]. A Paris, chez I'Autlieur,
demeurant en I'Isle du Palais, k I'lmage S. Michel. M.nc.xxvi. Auco priuilcgo
du Eoy. 8vo (paging begins at 341, and proceeds to 708). § 4.

Hensel. See HSlsse's Yeqa, § 4.

Hentsceek. Adrian Tlacq TabeUen der sinuum, tangentium . . . Ncue
und verbesserte Auflage von Johanit Jacob Uentschen-. Tranckfurt und
Leipzig, 1757. Small Svo (280 pp. of tables, 48 pp. of trigonometry, &c.).
§4.

Hekemann. ' Vienna Sitzungsberichte ' (Verbesserung der II. Callet'sohen
Tafel). See under Callet, 1853, § 4.

Hekwaet ab Hohenbtjrg. Tabulse arithmeticse Tlpoada^aipeereiits Uni-
versales, quarum subsidio numerus quilibet, ex multiplicatione producendus,
per Bolani additionem : et qnotiens quilibet, e divisione eliciendus, per solam
subtractionem, sine tsediosfi, & lubrica Multiplicationis, atque Divisionis ope-
ratione, etiam ab eo, qui Arithmetices non admodum sit gnarus, exacte,
celeriter & nuUo negotio invenitur. fe museo loannis Georgii Herwart ab
Hohenburg, V. I. doctoris, ex assessore summi tribunalis Imperatorii, et ex
Cancellario supremo serenissimi utriusque Bavarise Ducis, su£e serenissimse
Celsitudinis ConsUiarii ex intimis, Prsesidis provintise Schuabae, & inclytorum
utriusque Bavarise Statuum CanceUarii. Monachii Bavariarum, ex offlcina
Nicolai Henrici, Anno Christi m.dc.x. obi. folio (pp. 999 and introduction
7 pp.). § 3, art. 1.

Hill, John. Decimal and logarithmical Arithmetic explained . . . -with a
table of logarithms from 1 to 10,000 . . . Edinburgh, 1799. 8vo (pp. 46).
§ 3, art. 13.

HiNB, J. E, See [Faelet] (Versed Sines, 1856).

HoB"BHT, Jean Philippe and Loiris Idelek. Nouvelles Tables trigonome-
triques calcul^Es pour la ^vision decimale du quart de cerclo . . . Berlin,
1799. 8vo (pp. 351, and introduction Ixxii). § 4.

HoHENBUEs. See Herwart. ^

HotEL, J. Tables de Logarithmes h cinq d^cimalcs , . , Paris, 1858. 8vo
(116 pp. of tables, 32 of introduction). § 4.

HotTEi, J. Tables pour la reduction du temps en parties ddcimales du
jour . . . Publication der astronomischen GeseUschaft, iv. Leipzig, 1866.
4to (pp.27). §3, art. 12.

HtJLssE, J. A. See Vega (Sammlung, 1840).

HtJLSSE's Vega. See Vega (Sammlung, 1840.)

HuiioN (Tracts). § 2, art. 3.

HtJiTON, Chables. Tables of the Products and Powers of Numbers . . .
Published by the Commissioners of Longitude. London, 1781. folio Cpp.
•103). §4. ^^^

HuiTON, Charles. Mathematical Tables : containing common, hyperbolic,
and logistic logarithms. Also sines, tangents, secants, and versed sines . . .
to which is prefixed a large and original history of the discoveries and writings
relating to those subjects . . . London, 1785. 8vo (pp. 343 of tables and 170
of introduction). § 4 (under Huiioif, 1858).

HuiToir, Chaeles. A Philosophical and Mathematical Dictionary ... (in
2 vols.), vol. ii.- London, 1815. 4to. §3, art. 8.

HyiToir, Charles, Mathematical Tables, . . . with seven additional tables
o trigonometrical formulae by Olinihus Geegoei .. . New edition, London,
1858. 8vo (368 pp. of tables). §4.

Ideleb. See Cbntneeschweh.

Idelee. See Hobeet.

ON MATHEMATICAL TABLES. 153

Inman, J. Nautical Tables, designed for tlie use of Britisli Seamen. New
edition, revised by the Kev. J. W. Inman. London, Oxford and Cambridge,
1871 . 8vo (445 pp. of tables). § 4.

Iesbngakih, H. F. Gemeinnutziges Compendium von Quadrat-Flaohen-
Tabellen . . . Small 8vo. Hannover, 1810 (pp. 148 and xxxvi). § 4."

JiEGBK. See under KEtJGEit, § 3, art. 8.

Jahst, Gtjstav Adoipu. Tafeln der sechsstelligen Logarithmen fiir die
Zablen 1 bis 100 000, fur die Sinus und Tangenten . . . Leipzig. 2 vols,
vol. i. 1837; vol. ii. 1838. 4to (vol. i. pp. 79, and introduction, &c., xvi;
vol. ii. pp. 463, and introduction, &c., viii). There is also a Latin title on
the same titlepage. § 4.

JoNOoiniT, E. DE. De natura et prssclaro usu simplicissimee specie! nume-
rorum trigonalium . . . Hagas Comitum, 1762. Very small 4to (pp. 267).
§ 3, art. 25.

Jttngb, AtTGUST. Tafel der wirkliohen Lange der Sinus und Cosinus fiir
den Eadius 1 000 000 und fiir alle Winkel des ersten Quadranten von 10 zu
10 Secunden . . . . insbesondere fiir diejenigen, welche bei trigonometrischen
Berechnungen die Thomas'sche Eechenmaschine benutzen. Leipzig, 1864..
Small folio (pp. 90). § 3, art. 10.

KisTNEK (Geschichte der Mathematik). § 2, art. 3.

KjsrrH. See [Matnaed.]

Keplbe, J. Joannis Kepleri . . . Chilias logarithmorum ad, totidom nu-
meros rotundos . . . quibus nova traditur Arithmetica . . . Marpiirgi, 1624.
Small 4to (55 pp. of introduction and table unpaged). § 3, art. 18.

Keeigan, Thomas. The young Navigator's Guide to . . . Nautical Astro-
nomy . . . London, 1821. 8vo (204 pages of tables). § 4.

XlEBT. See BOWDITOH.

EoHLEE, H. G. Jerome de La Labde's logarithmische-trigonometrische
Tafeln durch die Tafel der Gausschen Logarithmen und andere Tafeln und
Formeln vermehrt . . . Stereotypen-Ausgabe. Dritter Plattenabdruck . . .
Leipzig, 1832. 32mo (pp. 254, and introduction xlv). There is also a
French titlepage. § 4.

KoHLEE, H. G. Logarithmisch-trigonometrisches Handbuch . . . Zweite
Stereo typausgabe. Leipzig, 1848. 8vo (pp. 388, and introduction xxxvi).

§4.

KeIjgee, J. G. Gedancken von der Algebra nebst den Primzahleu von 1.
bis 1 000 000 . . . Halle im Magdeburgischen, 1746. 12mo (Algebra pp. 124,
and the list of primes pp. 47). § 3, art. 8.

Ktjlik, Jakob Phtlipp. Tafeln der Quadrat- und Kubik-Zahlen aller
natiirlicheu Zahlen bis Hundert Tausend . . . nach einer ueuen Methode be-
rechnet . . . Leipzig, 1848. 8vo (pp. 460, and preface vii). § 3, art. 4.

Lalandb, , Jeeomb de. Tables de logarithmes pour les nombres et pour les-
sinus . . . Edition stereotype .... gravee, fondue et imprimee, par Firmin
Didot . . . Paris, 1805 (tirage de 1816). 16mo. § 4.

Laiande, Jeeome de. Tables de logarithmes par Jerome de Lalande eten-
dues h, sept deeimales par F. C. M. Maeie . . . precedees d'une instruction . . .
par le Baron Eeynaud. Edition stereotypee . . . Paris, 1829. 12mo (pp.
204 and introduction xlii). § 4.

Laiandb (Bibliographie Astronomique). § 2, art. 3.

Lalande. •See Kohlee (1832).

Lalande. See Ebtnaud.

Lambekt, J. H. Supplementa tabularum logarithmicarum et trigonome-
tricarum .... cum versioue introdutionis {sic), Germanicse in Latinum ser-

154 iiEi'OHT — 1873.

monem, Bocundum ultima auctoris consilia amplificata. Curante Antonio
Felkel. Olisipono, 1798. 8vo (pp. 198 and introduction Ixxv). § 4.

Lambebt, J. H. Zusatso zu den logarithmischen und trigonometrischen
Tabellen, 1770. See the Supphmenta ^c. of the same author next above,
§4.

Lattnbt, Samuel Linn. Table of Quarter-squares of all integer numbers
up to 100,000, by which the product of two factors may be found by tho aid
of Addition and Subtraction alone. . . London, 1856. 8vo (pp. 214 aud intro-
duction xxviii). § 3, art. 3.

LATJNBr, S. L. A Table of Products, by the factors 1 to Q of all numbers
from 1 to 100,000 . . . London, 1865. 4to (10 pp. of tables and introdiic-
tion vi). § 3, art. 1.

Lax, Eev. W. Tables to be used with the Nautical Almanac for finding
the latitude and longitude at sea . . . London, 1821. 8vo. § 4.

Lefoet, F. Description des grandes Tables logarithmiques ot trigonome-
triques calculees au Bureau du Cadastre, &c. Annales do I'Observatoire
Imperial de Paris, t. iv. (1858) pp. [123]-[150]. § 3, art. 13, under Tables
su Cadasibe.

Lbonelli. Leonelli's logarithmische Supplemente . . . aus dem Pranzo-
sischen nebst einigen Zusatzen von G. W. Leonhaedi . . . Dresden, 1806.
Small 8vo (pp. 88). § 3, ai-t. 19.

Leonhabdi. See Leonklli.

Leslie, John. The Philosophy of Arithmetic .... with tables for the
multiplication of numbers as far as one thousand . . . Second edition, im-
proved and enlarged. Edinburgh, 1820. 8vo (pp. 258). § 3, art. 3.
. LiTTEOw, C. L. VON. Hulfs-Tafeln fiir die Wiener Univorsitats-Sternwarte.
Zusammengestellt im Jahre 183/ . . . 8vo (pp. 88). § 3, art. 12.

Ltjdolf. Tetragonometria tabularia, qua per tabulas quadratorum b, Eadiee
qna;drata 1. usque ad 100 000 . . . Autore L. Jobo Ludolfeo, P. P. Math,
in Universitate Hierana ibidemque Senatore. Amstelodami, 1690. SmaU,
4to (introduction, 150 pp., and tables about 420 pp.). § 3, art. 4.

LnTN, Thomas. Horary tables, for finding the time by inspection . . ,
London, 1827. 4to (300 pp. of tables). § 4.

Mackat, Andeew. The Theory and Practice of finding the Longitude . . .
with new tables. In 2 vols., the third edition, improved and enlarged . . .
London, 1810. 8vo (vol. ii. contains about 340 pp. of tables). § 4,

Maqini, J. A. Tabula tetragonioa seu quadratorum numerorum cum suis
radicibus ex qua cujuscunque numeri perqnam magni minoris tamen triginta
tribus notis, quadrata radix facile, minimaque industria colligitur. Venetiis,
1592. § 3, art. 4.

Magints, J. a. . . . De Planis triangulis liber unicus. De dimetiendi
i-atione . . . libri quinque. Venetiis, 1592. SmaU 4to (contains the Tabula
Tetragonica, see Magini above). § 3, art. 4.

Maeie. See Lalanbe (1829).

Maekiott. See under Willich, § 4,

Maettn, C. F. Les tables de Martin, ou le r^gulateur universel ....
troisi^me Edition. Paris, 1801. 8vo. § 3, art. 1.

Maseees, Feancis. The Doctrine of Permutations and Combinations . , .
together with some other useful tracts . . . London, 1795. 8vo. § 4.

[Maskelxne, Nevil.] Tables requisite to be used with the Nautical Ephe-
meris . . . Published by order of the Commissioners of Longitude. The third
edition, corrected and improved. London, 1802, 8vo (206 pp. of tables, and
appendix (see next below) 106 pp. of tables). § 4. ■

ON MATilEJIATICAE T^ilSLES. 159

[M ASKELYNE, Netil.J Appendix to tte third edition of tlxe Eequisite Tables
... [London, 1802]. 8vo (pp. 106). § 4.

Maskelyke. See Michael Taxlob (1792).

Massaloot, J. V. Logarithmiacli-trigonometrische Hiilfstafein . . . Hand-
bucli fiir Geometer, Markscheidor ... Leipzig, 1847 (pp. 667 and intro-
duction xii). § 3, art. 10.

[Matthiessen, E. a.] Tafel zur bequemem Berechnung des Logarithmen
der Snmme oder Diflferenz zweyer Grossen welolie selbst nur durch ihre
Logarithmen gegeben sind. Altona, 1818. Large 8vo (pp. 212 and intro-
duetion 53). There is also a Latin titlepage. § 3, art. 19,

[Maynabd, Samuel.] A table containing useful numbers often required in
calculations, together with their logarithms. 8vo (pp. 12, numbered 169-
180). _ Prom Templeton's 'Millwright and Engineer's Pocket Companion?
[see title under Temi-leion]. It is stated on the first page that a portion of
tho table had appeared in other publications, and in particular in Keith's
' Measurer,' 24th edit. 1846, by the same editor (Maynard). § 3, art. 24.

Menboza. See liios.

Mebpaut, J, M. Tables Arithmonomiques fondles sur le rapport du reotr
angle au carr^ ou le calcul reduit k son dernier degre de simplification , . ,
Vannes, 1832. 16mo (500 pp. of tables, introduction 40 pp.), § 3, art. 3.

MioHABLis. See under Htjlsse's Yesa, § 4,

MiNsiNQEE, Prof. Die gemeinen oder Briggischen Logarithmen der Zahlen
. . . Augsburg, 1845. 8vo (31 pp. of tables and introduction &c. vi). § 4.

MoifiEBBEiEE, A. S. DE. Dictionnaire des sciences mathematiques pures et
appliquees . . . Tome troisieme (Supplement). Paris, 1840, foKo. § 3, art, 13i

Mokittcla (Histoire des Mathematiques). § 2, art. 3.

[MooEE, Sib Jonas.] A canon of the squares and cubes of all numbers
under 1000. Of the squared squares under 300. And of the square cubes
and cubed cubes under 200 . . . [London, 1650 ?] § 3, art. 4.

MooEB, SiK Jonas. Excellent Table for the finding the Periferies or Cir-
cumferences of aU. EUeipses or Ovals . , . (no place or date. ? London, 1660);
1 page foHo. § 3, art. 22.

MooEE, SiE Jonas. A new Systeme of the Mathematieks ... In 2 vols.
Vol. ii. (Tables). London, 1681. 4to (351 pp. of tables). § 4.

[MooEB, Sib Jonas.] A Table of Versed sines both natural and artificial.
4to. [London, 1681] (pp. 90). § 4.

MooBE, J. H. The new Practical Navigator ; being a complete epitome
of navigation, to which are added aU. the Tables requisite . . , The nineteenth
edition, enlarged and carefully improved by Joseph Dessioxt. London, 1814.
8vo. § 4.

Moitton's sines &o. to every second. See Gaedinee (Avignon reprint, 1770),

MUliee, J. H. T. Vierstellige Logarithmen der natiirlichen Zahlen und
Winkel Eunctionem . . . (Preface dated from Gotha, 1844.) 8vo (25 pp. of
tables). § 4.

*Mtjltiplication, Tables, de . . . Paris, 1812. § 3, art. 1 (Introductory
remarks).

Muehaeii (Bibliotheca Mathematica). § 2, art. 3.

Naplee. Mirifici Logarithmomm Canonis de^criptio, Ejusque usus, in
utraque Trigonometria ; ut etiam in omni Logistica Mathematica, Amplissimi,
EaciUimi, & expeditissimi explicatio. Authore ac Inventore, Ioanne Nepeeo,
Barone Merchistonii, &c. Sooto. Edinburgi, Ex officina Andreae Hart Bib-
liopofe, cio.DC.xiv. [On an ornamented titlepage.] 4to (dedication, preface
&c. 6 pp., text 67 pp., tables 90 pp.). § 3, art. 17.

156 REPORT — 1873.

Napieh. Mirifici logarithmorum canonis constructio ; Et eorum ad natu-
rales ipsorum numeros habitudines ; uuJk cum Appendioe, de alia eaqu&
praestantiore Logarithmorum specie condenda. Quibua acoessere Proposi-
tiones ad triangula sphrerica faciliore calculo rcsolvenda : JJnk cum Anno-
tationibus aliquot dootissimi D. Hcnriei Briggii, in eas & memoratam appen-
dicem. Authoro & Inventore loanne Nepero, Barone Merobistonii, &e.
Scoto. [Typographical ornament, a thistle.] Edinburgi, Excudebat Andreas
Hart. Anno Domini 1619. 4to (preface 2 pp. and text 67 pp.). § 3, art. 17.

[The above is a transcript of the titlepage of the 'Constructio;' but in the only
copy of thiswork that we have seen it is immediately preceded by an ornamental
titlepage, whioh, as far as the ornamentation is concerned, is a facsimile oS that
of the ' Descriptio,' 1614. The letterpress, however, is very different, and runs,
^' Mirifici logarithmorum canonis descriptio, Ejusque usus, in utraque Tri- .
gonometria ; ut etiam in omni Logistica Mathematica, amplissimi, facillimi,
<k expeditissimi explicatio. Accesserunt opera posthuma : Prim6, Mirifici
ipsius canonis constructio, & Logarithmorum ad naturales ipsorum numeros
habitudines. Secundo, Appendix de alia, eftque prasstantiore Logarithmorum
specie construenda. Tertid, Propositiones quiedam eminentissimss, ad Tri-
angula spbaerica mirS, facilitate resolvenda. Autore ac Inventore loanne
Nepero, Baxone Merchistonii, &c. Scoto. Edinburgi, Excudebat Andreas
Hart. Anno 1619." This would imply that the ' Descriptio' and ' Constructio'
were issued together in 1619 ; and whether this was so or not, it shows that
such was intended. Some writers speak of a reprint of the ' Descriptio ' in
1619 ; but this title may be all their authority, as few of those who have
written on the subject seem to have looked beyond the titlepages of the
works they were noticing. On the other hand, of course, the 'Descriptio' may
have been torn out from the copy before us. The ' Constructio' is a much
rarer work than the 'Descriptio ;' we- have seen half a dozen copies of the
latter and but one of the former (Camb. Univ. Lib.). In any case, as
the leading words of the title of the ' Constructio' (on the first titlepage) are
." Mirifici logarithmorum canonis descriptio," it could only be distinguished
from the ' Descriptio ' in most library catalogues by the date 1619. We have
thought it worth while, since the description in § 3, art. 17 (p. 73), was
printed, to add the first title of the work containing the ' Constructio,' and to
point out the uncertainty relating to the reprint of the ' iDescriptio,' in hopes
that some one may settle the matter. The 1619 edition of the ' Descriptio'
(supposing there to have been one of this date) is the only book of importance
relating to the early spread of logarithms of which we have seen no copy;
and the question of its publication is almost the only point of bibliography,
in reference to the tables of this time, that we are obliged to leave undecided
for the present.]

Nepek, Nepair, or Nepper. See Napier.

Newton, John. Trigonometria Britanica (sic) : or, the doctrine of tri-
angles. In Two Books. . . . The one Composed, the other Translated, from
the Latine Copie written by Henry Gellibrand, ... A table of logarithms
to 100.000, thereto annexed, With the Artificial Sines and Tangents, to the
hundred part of every Degree ; and the three first Degrees to a thousand
parts. By John Newton . . . London : MDCLVIII. fol. (Dedication and
preface 6 pp., trigonometry 96 pp. ; tables unpaged.) § 4.

NoBiE, J. W. A complete set of Nautical Tables containing all that are
requisite . . . Eighth (stereotype) edition. London, 1836. 8vo (360 pp. of
tables). § 4.

NoBiE, J. W. A complete epitome of Practical Navigation , . . Thirteenth

ON MATHEMATICAL TABLES. 157

(stereotype) edition, considerably augmented and improved. London, 1844.
8yo (360 pp. of tables). §4.

[We have also seen the " fourteenth (stereotype) edition by George

Coleman," 1848, the "twelfth (stereotype) edition," 1839, the " eleventh
edition," 1835, all containing 360 pp. of tables — and, besides, an edition of
1805 containing 252 pp. of tables, in which it is stated that the tables were
published two years previously under the title " Nautical Tables."]

NoEWooD, EiCHARD. Trigonometric, or the Doctrine of Triangles . . . per-
formed by that late and excellent invention of logarithms . . . London, 1631,
Small 4to. § 4.

. Oakes, Lieut.-Col. W. H. Table of the reciprocals of numbers from 1 to
100,000, with their differences, by which the reciprocals of numbers may be
obtained up to 10,000,000. . . London, 1865. 8vo (205 pp. of tables and xii
of introduction). § 3, art. 7.

Oakes. Machine table for determining primes and the least factors of
composite numbers up to 100,000. Dedicated, by permission, to Professor
De Morgan. By Lieut.-Col. W. H. Oakes. Printed and published by
Charles and Edwin Lay ton. . . . London, 1865. § 3, art. 8.

Oppoxzek, Theodoe. Vierstellige logarithmisch-trigonometrische Tafeln.
. . . Wien, 1866 (pp. 16). § 4.
Opus Paiaxinitm. See Rheticus.
Otho. See Eheiicus (Opus Palatinum).

Ou&HTEED, William:. Trigonometrie, or. The manner of calculating tho
Sides and Angles of Triangles, by the Mathematical Canon, demonstrated . . .
published by Eichard Stokes and Arthur Haughton .... London, 1667.
Small 4to. (Trigonometry 36 pp., tables 240 pp.). § 4.

OzANAM, M. Tables des sinus tangentes et secantes et des logarithmes des
sinus et des tangentes . . . Paris, 1685. Small 8vo. § 4.

Pakkhitrsi. Astronomical Tables, comprising logarithms from 3 to 100
decimal places, and other useful Tables. By Henet M. Paekhuest. Eevised
edition. Printed and published by Henry M. Parkhurst (Short Hand Writer
and Law Eeporter), No. 121 Nassau Street, New York City. 1871. 12mo
(176 pp. of tables, 66 pp. of formulae, explanations, &c.). § 4.

Pasqfich, Joannes. Tabulae logarithmico-trigonometricae contractse cum
novis accessionibiis . . . Lipsiae, 1817. 8vo (pp.228 and introduction xxxviii).
There is also a German titlepage. § 4.
, Peacock (Arithmetic). § 2, art. 3.

Peaesoit, W. An introduction to Practical Astronomy containing Tables
.... London, 1824. 2 vols. Large 4to. § 4.

[Pell, J.] Tabula Numerorum Quadratorum decies millium, una cum ip-
sorum lateribus ab unitate incipientibus & ordine naturali usque ad 10 000
progredientibus . . . London, 1672. 4to (pp. 32). § 3, art. 4.

Petees, C. P. W. Astronomische Tafeln und Pormeln. . . Hamburg, 1871.
8vo(pp. 217). §4.

PEzEjfAS. See Gaedinee (Avignon edition, 1770).

Phillips, Sie Thomas, Bart. An improved Numeration Table to facilitate
and extend Astronomical Calculations . . . [London?], 1829. 12mo (pp. 18).
§ 3, art.. 25.

PicAETB, E. La Division reduite jt una Addition, ouvrage approuve par
I'Academie des Sciences de Paris . . . augmente d'une Table de Logarithmes
, . . Paris [1861]. 4to (pp. 104 and introduction &c. xvi). § 3, art. 7.

PiGEi, GiirsEPPE. Nuove Tavole degU Elementi dei Numeri dall' 1 al
10 000 .. . Pisa, 1758. 8vo (pp. 195). § 3, art. 8.

158 REPOKT— 1873.

Pdteto, S. Tables de Logarithmes vulgaires h dix d^cimaJes coiistniites
d'apres iin nouveau mode . . . S.-Petersbourg, 1871. 8vo (pp., 56 and intro-
daction xxiv). § 3, art. 13.

Puisctrs. Thesaurus BiathematicuB Sive canon sinuum ad radium
1.00000.00000.00000. et ad dena qusequc scrupula secunda Quadrantis:
una cum sinibus primi et postremi gTadus, ad oundem radiuin, et ad singula
scrupula secunda Quadrantis : Adjunclis ubiquo difFerentiis primis ot secuiir
dis; atq, ubi res tulit, etiam tertijs. jam olim quidem incredibili labore &
sumptu k Georgio Joachimo Ehetico supputatus : at nunc primum in lucCm
editus & cum yiris doctis commimicatus a Bartholomseo Pitiaco Gruubergensi
Silesio. oujuB etiam accesserunt : I. Principia Sinuum, ad radium, 1.00000.
00000.00000.00000.00000. qu&m accuratissim^ supputata. II. Sinus deci-
morum, tricesimorum & quinquagesimorum quorumq; scrupulorum secundo-
rumperprima &postrema35. scrupula prima, ad radium, 1.00000.00000.00000,
00000.00. [Typographical ornament.] Francofurti Excudebat Nicolaus
Hoffinannus, sumptibus Jonas Eosse Anno cio. lo. xiii. folio [part of the title
is printed in red] (preface 5 pp., tables pp. 2-271, pp. 2-61, pp. 3-15). There
are four titlepages altogether, including that to the whole •work (copied
above) ; on the first two the date should be cio. loc. xin, and not as printed.
§ 3, art. 10.

PoGGENDOEFF (Haudworterbuch). § 2, art. 3.

Pbasse. See De Pbasse.

Pront. See Tables dtt Cadastee. See also § 3, art. 13 (introductory
remarks, p. 54), and § 3, art. 16 (introductory remarks, p. 69). ^

Eahn, J. H. Teutsohe Algebra, oder Algebraische Eechenkunst . . . Zurich;
1659. Very small quarto (pp. about 200). § 3, art. 8.

EankIne, W. J. M. Useful Eules and Tables relating to Mensuration,
Engineering, Structures, and Machines . . . London,- 1866. 8vo. § 4. '

Eaper, IIenet, Lieut. E.N. Tables of logarithms to six places . . . London,
1846. 8vo (pp. 122 and introduction xi). § 4.

Eapeb, Henet, Lieut. E.N. The Practise of Navigation and Nautical
Astronomy . . . Sixth Edition. London, 1857. 8vo (454 pp. of tables). § 4.

Eees, Abraham. The Cyclopaedia, or Universal Dictionary of Arts,
Sciences, and Literature ... In 39 vols. London, 1819. 4to. Vol. xviii.
Hyperbolic logarithms. § 3, art. 16. Vol. xxi. Logarithms. § 3, art. 13;
Vol. xxviii. Prime numbers. § 3, art. 8.

Eeishammee, EiiLix. Manuel general pour les Arbitrages de CJhanges • . .
par Nomhres fixes ou par Logarithmes . . . suivi d'une Table de Logarithtoes
depuis 1 jusqu'i 10400 (et, k I'alde de la Tables des Differences, jusqu'i
104000) . . . Paris. An viii (1800). 8vo (pp. 326 and 131 pp. of 'tables).
§3, art. 13.

EEainsiiE Tables. See [MASKEirira;.]

Eetjss (Eepertorium), § 2, art. 3.

Ebtwaud, a. a. L. Trigonom^trie . . . troisieme ddition ; suivie des tables
de logarithmes , . . de Jer6me do Lalande. Paris, 12mo, 1818 (203 pp. of
tables). § 4,
• Eetnattd. See Lalakbe (1829).-

Ehetiotjs. Opus Palatinum de triangulis a Georgio loachimo Ehetico
coeptum: L. Valentinus Otho Principis Palatini Friderici IV. Electoris
mathematicus consummavit. An. sal. hum. cio. lo. xcvi, Plin. lib. xxxvi.
Oap. ix, Eerum naturae interpretationem ^gyptiorum opera philosophim
continent. Cum privilegio cses. majes. folio, 2 vols, [on an ornamented title-
page]. §3, art. 10.

ON MATHEMATICAL TABLES. 159

Ehetictjs. See PiiiscOT.

EiDDLB, Edwaeb. Treatise on Navigation and Nautical Astronomy . . '.
with all tte Tables, requisite in nautical computations . . . London, 1824.
8vo (239 pp. of tables). §4.

' Riiey's Arithmetical Tables for multiplying and dividing sums to the
utniost extent of numbers . . , London, 1775. 8vo (pp. 170 and intro-
duction xii). § 3, art. 1.

Eios, Joseph de Mendoza. A complete collection of Tables for Navigation
and Nautical Astronomy . , . Second edition, improved. London, 1809, '
4to (604 pp. of tables). § 4.

Rios, Josi: DE Mejstdoza t. Colecoion completa de Tablas para los uses de
la. Navegacion y Astronomia N^utica . . , Primera Tirada. Madridj 1860.
4to. §4.

Eoe, N. Tabulse Logarithmiese, or two tables of logarithmes ... by Na-
THANiEi, Eoe, Pastor of Benacre in Suffolke . . . Unto which is annexed their
admirable use ... by Ebm. Winsate, Gent. London, 1633. 8vo (preface and
tables unpaged, the Use &c. pp. 70, and 10 addit. pp. of tables). § 4.

EoGG (Bibliotheca Mathematica). §.2, art. 3.

EosENBEEa. See Dase (ninth niillion).

EoirsE, WiiiiAM. The Doctrine of Chances, or the Theory of Gaming
made easy . . . with Tables on Chance, never before published * . , London
[no date]. 8vo (pp. 350, preface &c. Ivi). § 3, art. 25.

EtfMEBE,. G. Handbuch der Schifffahrtsiunde mit einor Sammlung von
Seemanns-Tafeln , . . Vierte Auflage. Hamburg, 1844. 8yo (531 pp. of
tables). § 4.

Saigey. See under Cailet, 1858, § 4.

*Saxomoh-, Jos. M. Logarithmische Tafeln, euthaltend die Logarithmett
der Zahlen 1-10800, die Logarithmeh der Sinusse und Tangenten von
Sekunde zu Sekunde, etc. "Wien, 1827. .4to (pp. 466 and introduction
xxxviii). Also with Prench text. § 4.

Sang, Ebwaeb. Pive-place logarithms . . . Edinburgh and London, 1859.
32mo (pp. 32). § 3, art. 13.

Sajtg, Ebwaeb. A new table of seven-place logarithms of ail numbers from
20 000 to 200 000 .. , . London, 1871. Large 8vo (pp. 365). § 3, art. 13*

Sang, Ebwaeb. ' Edinburgh Transactions,' vol. xxvi. 1871. (Account of
the new table of logarithms to 200 000). See under Sang, § 3, art. 13.

ScHBiBEL (Mathematical Bibliography). § 2, art. 3.

[ScHEiTTz, G. and E.] Specimens of Tables ; calculated, stereomoulded,
and printed by Machinery. London, 1857. 8vo (pp. 50). § 3, art. 13.

*ScHioMncH, 0. Fiinfstellige logarithmische und trigonometrischo Tafeln.
Braunschweig. 8vo. § 4.

Schmidt, G. G. Logarithmische, trigonometrisehe und andere Tafeln
. . . Giesseh, 1821. 12mo (pp. 217 and introduction xxii). § 4.

ScHEON, LtTBWiG. Tafeln der drei= und fiinfstelligen Logarithmen . . . Jena,
1838. (Small quarto tract, without cover, 20 pp.) § 3, art. 13.

ScHEON, Lttbwig. Siebenstellige gemeine Logarithmen der Zahlen von
1 bis 108000 und der Sinus, Cosinus, Tangenten und Cotangenten . . . nebst
einer Interpolationstafel zur Berechnung der Proportionaltheile . . . Stereo-
typ-Ausgabe. Gesammt-Ausgabe in drei Tafeln. Braunschweig, 1860. Large
8vo (pp. 550). § 4.

SoHEON, LrnwiG. Seven-figure logarithms . . . Fifth edition, corrected
and stereotyped. With a description of the tables addedby A. db Moegan . . .
London and Brunswick, 1865. Svo. § 4.

160 repout— 1873.

ScHTJLZE, JoHANN Carl. Ncug und erwciterte Sammlung logarithmischer,
•trigonometrischer und anderer. . . .Tafeln. Berlin, 1778. 2 vols. 8vo (each
about 300 pp.). There is also a French titlepage. § 4.

SCHUIZE. See ACADfiMIE EOYALB DB Pkusse, § 4.

ScHUMACHEE, H. Gr. Sammlung von Hiilfstafeln herausgegeben im Jahre
1822 von H. Or. Schumacher. Neu herausgegeben und vermehrt von G. H.
L. Warnstoevf. Altona, 1845. 8vo (pp. 221, and 31 pp. of explanation in
French). § 4.

[Schumacher.] Auxiliary Tables for Mr. Bessel's method of clearing the
Distances. Svo (pp. 91). [No editor's name, date, or place.] § 4.

Schweigsee-Seidel (Litteratur der Mathematik). § 2, art. 3.
■ SioxriN, M. Manuel d' Architecture ou Principes des Operations primi-
tives de cet Art .... Get ouvrage est terming par une table des quarries et des
cubes, dout les racines commencent par Tunit^, et vont jusqu'ik dix mille ....
Paris, 1786. 8vo (the table occupies 100 pp.). § 3, art. 4.

Shanks, "William. Contributions to Mathematics, comprising chiefly the
Eectification of the Circle to 607 places of decimals . . . London, 1853. Printed
for the Author. Svo (pp. 95). § 4.

[Sharp, Abraham.] Geometry Improv'd. 1. By a large and and accurate
table of segments of circles. . . .with compendious tables for finding a true
proportional part . . . exemplify'd in making out Logarithms or natural numbers
from them, true to sixty figures, there being a table of them for all primes to
1100, true to 61 figures. 2. A concise treatise of Polyedra. ... By A. S.
Philomath London, 1717. Small 4to (pp. 136). § 4.

Sharp. See Sheewist.

Sheepshanks, E. Tables for facilitating Astronomical Eeductions. London,
1846. 4to. § 4. (Also Anontmoto, 1844). § 4.

[Shbrwin, Henry.] Sherwin's Mathematical Tables, contriv'd after a
most comprehensive method .... The third edition. Carefully revised and
corrected by William Gardiner. London, 1741. 8vo. § 4.
• Shortrede, Eobeet. Compendious Logarithmic Tables .... Edinburgh,
1844.- Svo (pp. 10). § 4.

Shoetrede, Eobeet. Logarithmic Tables to seven places of decimals

containing Edinburgh, 1844. Large Svo (pp. 829, and introduction,

pp. 39). § 4. Also 1849 (2 vols.). See next title.

Shoetrede, Eobeet. Logarithmic Tables : containing logarithms to num-
bers from 1 to 120,000, numbers to logarithms from -0 to 1-00000, to seven
places of decimals ; . . . . Edinburgh, 1849. Svo (pp. 208 and preface xxv).
This is the title of the first volume; that of the second is, •' Logarithmic
Tables to seven places of decimals, containing logarithmic sines and tan-
gents to every second of the circle, with arguments in space and time ..."
Edinburgh, 186S (pp. 602 and preface pp. 2), Svo. The two volumes soem
to have been regarded as separate works, as the book is not stated to be in
2 vols ; nor are they called vol. i. and vol. ii. § 4, under Shoetrede, 1849.

SoHNKE (Bibliotheca Mathematiea). § 2, art. 3.

Speidell, J. New logarithmes. the First inuention whereof, was, by the
Honourable Lo : lohn Nepair Baron of Marchiston, and Printed at Edinburg
in Scotland, Anno : 1614. In whose vse was and is required the knowledge
of Algebraicall Addition and Subtraction, according to + and — These being
Extracted from and out of them (they being first ouor scene, corrected, and
amended) require not at all any skiU. in Algebra, or Cossike numbers, But
jnay be vsed by euery one that can onely adde and Subtract, in whole numbers,
according to the Common or vulgar Arithmeticke, without any consideration

ON MATHEMATICAL TABLES. 161

or respect of + and — [Typographical ornament] By lohn Speidell, pro-
fessor of tte Mathematickes ; and are to bee soldo at his dwelling house in
the Fields, on the backe side of Drury Lane, betweene Princes streete and the
new Playhouse. [Erasure in ink.] 1619 (unpaged, pp. 90 and tiUepage).
§ 3, art. 16.

Stansbtjrt, Daniel. Tables to facilitate the necessary Calculations in

Nautical Astronomy New York, 1822. 4to (337 pp. of tables). § 4.

[S'lBG-MAifir, F.] Tafel der fiinfstelligen Logarithmen und Antilogarithmen.
Marburg, 1855. § 4.

*SxE6MAmr. Tafel der natiirlicher Logarithmen. Marburg, 1856. § 4.
SiBiNBEESEE, A. Tafel der gemeinen oder Brigg'schen Logarithmen aller
Zahlen von 1—1 000 000 mit fiinf und beliebig sieben Decimalstellen ....
Eegensburg, 1840. 8vo (pp. 65). § 3, art. 13.

Tabibs dv Cadasiee, calculated under the direction of Pkont (manu-
script). § 3, art. 13.

Tatxoe, Janbt. Lunisolar and Horary Tables, with their application in
Nautical Astronomy. . . . London, 1833. 8vo (pp. 232). § 4.

Tayloe, Janet. An Epitome of Navigation and Nautical Astronomy,
with the improved Lunar Tables .... London, 1843. 8vo (320 pp. of
tables). § 4.

Taxloe, Michaei,. a Sexagesimal Table .... and the Sexagesimal Table
turned into seconds as far as the 1000th column. . . . Published by order of
the Commissioners of Longitude. London, 1780. 4to (pp. 316 and intro-
duction xlv) § 3, art. 9.

Tatloe, Michael. Tables of logarithms of AU numbers, from 1 to 101000,
and of the sines and tangents to every second of the quadrant. . . . With
a preface. . . .by Nevil Maskelyne. . . . London, 1792. Large 4to (about
600 pp.). § 4.

Tempieion, W. The Millwright and Engineer's pocket Companion . . .
corrected by Samuel Maynard : London, 1871. 8vo. (Noticed under [May-
naed], § 3, art. 24).

Thomson, Davtd. Lunar and Horary Tables .... Forty-fourth edition.
London, 1852. 8vo (218 pp. of tables). § 4.

Todd, Chaelbs. A series of Tables of the Area and Circumference of
Circles ; the Solidity and Superficies of Spheres ; the Area and Length of the
Diagonal of Squares.... Second edition. London, 1853. 8vo (pp. 114).
§ 3, art. 22.

Teoitee, James. A Manual of Logarithms and Practical Mathematics ....
Edinburgh, 1841. 8vo (82 pp. of tables). § 4.

TuEKisH Table of Logarithms &c. [Bul^k] 1250 [1834]. 8vo (pp. 270).
§4.

Uesiw. See G. F. Uesinits.

Uesinits, B. Beni. Ursini Mathematici Electoralis Brandenburgici Trigo-
nometria cum magno logarithmor. Canone Cum Privilegio Colonise Sumptib.
M. Guttij. tipijs G. Rungij descripta CD DCXXV (sic). (This is the title of
the volume, and is printed on an ornamented titlepage.) The trigonometria
occupies 272 pp. ; and then follows the Canon, unpaged, with a fresh title-
page. "Benjaminis Ursini Spottavi Silesi .... Magnus Canon triangulorum
logarithmicus ; ex vote & consilio Illustr. Neperi, p. m. novissimo, Et sinu
toto 100,000,000. ad scrupulor. secundor. decadas usq; vigili studio & perti-
naci industria diductus . . . Colonise. Typis Georgij Kungij . . . M.DC.XXIV";
but the colophon (at the end of the canon and of the whole work) is
" Berolini, Excudebat Georgius Ktmgius Typographus, impensis & sumtibus
1873. M

162 iiEPORT— 1873.

Martini Guttij. Bibliopolse Coloniensis. Anno Clg IgC XXIV." 4to. § 3,
art. 17.

Ursinus, G. F. Logarithmi VI Decimalium scilicet numerorum ab 1 ad
100 000 et Sinuum et Tangentium ad 10" . . . (Impensis autoris.) Hafnise,
1827. 8vo. §4.

Veba, G. Thesaurus logaritlimorum completus, ex arithmetica logarithmica,
et ex trigonometria artficiali Adriani Vlacci coUectus, plurimis erroribus
purgatus, in novum ordinem redactus, .... Wolframii denique tabula logarith-
morum naturalium locupletatus a Georgio Vega .... Lipsise, 1794. folio
(pp. 685 and introduction xxx). Tkere is also a German titlepage. § 4.

Vega, G. Georgii Vega .... tabulae logarithmico-trigouometricae cum
diyersis aliis in Matheseos usum constructis Tabulis et Pormulis .... Editio
secunda, emendata, aucta penitusque reform ata. LipsiiB, 1797. 2 vols. 8vo
(pp. 409 and 371 ; vol. i. has also Ixxxiv pp. introduction). There is also a
German titlepage. § 4.

Vega, G. Georgii Vega . . . . manuale logarithmico-trigonometricum. . . .
Editio secunda, aucta et emendata. Lipsise, 1800. 8vo (pp. 304 and intro-
duction bciv). There is also a German titlepage. § 4.

Vega, G. Sammlung mathematiseher Tafeln. . . . Herausgegeben von Dr.
J. A. HtJLSSE. Stereotyp-Ausgabe. Erster Abdruck. Leipzig, 1840. 8vo
(pp. 681 and introduction xxiv). § 4 (described as HSlsse's Veua).

Vega, G. Logaritimiisch-trigonometrisches Handbuoh (einundvierzigste
Auflage) .... bearbeitet von Dr. C. Bremikek. Berlin, 1857. 8vo (pp. 575
and introduction xxxii). § 4 (described as Bbemikek's Vega).

V.EGA, G. Logarithmic Tables by Baron von Vega, translated from

the fortieth edition of Dr. Bremiker's by W. L. F. Fischer .... Thoroughly
revised and enlarged edition .... .Stereotyped .... Berlin, 1857. (pp. 575 and
introduction xxvii) § 4 (under Bremiker's Vega).
Versed Sines, A Table of. See [Sir Jonas Moore.]
Vbrseb, Sines, Natural . ... and Logarithmic . . . See [Fablet].
Vlacq, Adrian. Arithmetica logarithmica, sive logarithmorum chiliades
centum, pro !5fumeris naturiili serie crescentibus ab Unitate ad 100000.
una cum canone triangulorum seu tabula artiflcialium Siauum, Tangentium,
& Secantium, Ad Radium 10,00000,00000. & ad singula Scrupula Prima Qua-
drantis. Quibus novum traditur compendium, quo nuEum nee admirabHius,
nee uliUus solvendi pleraque Problemata Arithmetica & Geometrica. Hos
numeroB primus invenit Clarissimus Vir lohannes Neperus Baro Merchis-
tonij : >eos autem ex ejusdem sententia mutavit, eorumque ortum & usum
iUustravit Henricus Briggius, in celeberrimft Academia Oxoniensi Geometrise
Professor Savilianus. Editio Secunda aucta per Adrianum Vlacq .Goudanum.
Deus nobis usuram vitae dedit et ingenii, tanquam pecuniae, nidla praestituta
die. [Typographical ornament.] Goudae, Excudebat Petrus Eammasenius.
M.DC.XXVm. Cum PrivUegio Illust. Ord. Generalium. fol. (preface and
errata 5 pp., trigonometry &c. 79 pp.; tables unpaged). Part of the title is
printed in red. § 4.

ViAca, Adrian. Arithmetique logarithmique ou la construction et usage
d'une table contenantlesLogarithmesde tousleaNombresdepuis I'TJnite jusques
A 100000. et d'une autre table en laquelle aont comprins les Logarithmes des
Sinus, Tangentes & Secantes, de tons les Degrez & Minutes du quart du
€ercle, selon le Raid de 10,00000,00000. parties. Par le moyen desquelles
«n resoult tres-facilement les Problemes Arithmetiques & Geometriques.
Ces n ombres premierement sent inventez par lean Neper Baron de Mar-
chiston; mais Henry Brigs Professeur de la Geometrie en I'Universite

ON MATHEMATICAL TABLES. 163

d'Oxford, les a change, & leur Nature, Origine, & Usage illustre selon I'inten-
tion du dit Neper. La description est traduite du Latin en Frangois, la
pi-emiere Table augmentee, & la seconde oomposee par Adriaen Vlacq. Dieu
nous a donne I'usage de la vie et d'entendement, plus qu'il n'a fait par le
temps passe. [Small typographical ornamentl. A Goude, Chez Pierre
Sammasein. M.DC.XXVIII. Aveo Privilege' des Estats Generaux. fol.
(preface 3 pp., errata 1 p., trigonometry &o. 84 pp. ; tables unpaged). Part
of the title is printed in red. § 4.

[The radius is erroneously describedin the above two titles as 10,00000,00000;
it is really 1,00000,00000, viz. the logarithms are given to ten decimal places.]

Vlacq, Adeiast. Logarithmicall arithmetike. or tables of logarithmes for
absolute numbers from an unite to 100000 ; as also for Sines, Tangentes
and Secantes for every Minute of a Quadrant; with a plaine description of
their use in Arithmetike, Geometrie, Geographie, Astronomie, Navigation,
&o. These Numbers were first invented by the most excellent lohn Neper
Baron of Marchiston, and the same were transformed, and the foundation
and use of them Ulustrated with his approbation by Henry Briggs Sir Henry
Savils Professor of Geometric in the Universitie of Oxford. The uses
whereof were written in Latin by the Author himselfe, and since his death
published in En^ish by diverse of his friends according to his mind, for the
benefit of such as understand not the Latin tongue. Deus iiobis usuram
vitse dedit, et ingenii, tanquam pecuniae, nulla prsestituta die. [Printer's
device and motto, Anehora spei.] London, Printed by George Miller.' 1631,
fol. (54 pp. of trigonometry &c. followed by "a Table of Latitudes" (8 pp.),
ajid then the logarithmic tables, unpaged). § 4.

Ylacq, Adkian. Trigonometria artiflcialis : sive magnus canon triangu^
lorum logarithmicus. Ad Eadium 100000,00000, & ad dena Scmpula Secunda,
ab Adriano Vlacco Goudano Constructus. Cui Accedunt Henrici Briggii
Geometriae Professoris in Academia Oxoniensi p.m. Chiliades logarithmorum
Viginti pro numeris naturali serie cresoentibus ab Unitate ad 20000. Quorum
ope triangula plana & sphserica, inter alia Nova eximiaque compendia e
Geometricis fundamentis petifca, sola Additione, Subtractione, & Sipartitione,
exquisitissime dimetiuntur, [Here follows a quotation of seven fines from
Kepler. Harm. lib. iv. cap. vii, p. 168,] Goudse, Excudebat Petrus Bam^
masenius. Anno M.DCXXXIII. Cum Previlegio. folio. (Dedication and
preface 4 pp., trigonometry &c. 52 pp. ; tables unpaged). § 4.

VLAca, Adbiaw. Tabulffi sinuum, tangentium et logarithmi sinuum tangen,-
tium & numerorum ab unitate in 10,000 . , . . Editio ultima emendata &
aucta. Amstelscdami : Apud Henricum & Viduam Theodori Boom. 1681,
Small Svo. § 4.

Viaoq's works (Chinese reprint). § 3, art. 13 (introductory remarks, p. 54).

ViAca. See HBUTscHEif.

*VoisnT, AwToijJB. Tables de Multiplications ou Logarithmes des Nombres
Entiers depuis 1 jusqu'a 20,000 Paris, 1817. § 3, art. 3. .

Wacsekbaeth, a. E. Di Fem-sttlliga Logarithm -Tabeller, jemte en
SanOing Tabeller Upsala, 1867. Small 8vo (pp. 224 and introduction

xviii). § 4.

Waxlacb, John. Mathematical Tables containing the logarithms of num-
bers, logarithmic sines, tangents, and secants By J. Brown. The third

edition, improved, enlarged with many useful additions, by J. Wallacj;,
Edinburgh, 1815. Svo. § 4.

Wallis. See Shbewin.

Waejtstobff. See Schumaches,

M 2

164 REPORT— 1873.

Weidenbach. Tafel um den Logarithmen von — -r zn finden wenn der

Logarithme von x gegeben ist . . . . Mit einem Vorworte von Herrn Hofrath.
Gattss. Copenhagen, 1829. 16mo (pp. 24). § 3, art. 19.

"Wells, I. Sciogiaphia. London, 1635. See under De Deckbe, 1626.

WiLLicH, C. M. Popular Tables arranged in a now form Third edition.

London, 1853. Svo (pp. 166). § 4.

WiNGATE. See EOE.

WiTTSTEiN, THEoroE. Logarithmes de Gauss & sept ddcimales .... Han-
nover, 1866. 8vo (pp. 127 and introduction xvi). § 3, art. 19.

■WotFEAM. 48-place hyperbolic logarithms: these first appeared in ScmrizE's
Sammlung. See Schtjlze (1778).

"WooLHousE, W. S. B. On Interpolation, Summation, and the Adjustment
of Numerical Tables London, 1865. Svo (pp. 100). § 3, art. 21.

WooLHOusE. See Olinthtcs Geegory (1843).

■WucHBKER, W. F. Beytrage zum allgemeinem Gebrauch der Decimal-
Briiche Carlsruhe, 1796. Svo (152 pp. of tables and 48 pp. of intro-
duction). § 3, art. 6.

Zech, J. Tafeln der Additions- und Subtractionslogarithmen fiir sieben
Stellen .... Aus der Vega-Hiilsse'sohen Sammlung besonders abgedruckt.
Leipzig, 1849. Svo (pp. 201). Also " Zweiter Auflage," 1863. § 3, art. 19.

§ 6. Postscript.

Art. 1. The foregoing Report is that which was presented to the Brighton
Meeting in 1872, considerably enlarged. After the Meeting it seemed de-
sirable to extend some of the articles in § 3, and to add descriptions of several
works to § 4 ; and it then appeared that the Report was so lengthy that it
was thought better to delay its publication till the ensuing volume, so as to
afford time for its passage through the press without undue haste. The
printing therefore was commenced in February or March, and is now
(September 30, 1873) all but finished. It was arranged, as the completion
of the Report by a supplement depended in great measure on the coopera-
tion of others possessing information on the subject of tables, that a certain
number of separate copies should be placed in the bands of the Committee,
as soon as the printing was effected, for circulation amongst those interested
in the matter, so as to avoid the delay of a year that would otherwise take
place before the work undertaken by the Committee became known to those
who could render assistance.

Art. 2. While the Report has been passing through the press a good many
alterations have been made which were necessitated by increased informa-
tion on the subjects treated of, and by repetitions &c. which were detected
for the first time when the whole appeared in print. But no attempt has
been mads to increase the extent of the Report by introducing descriptions
of fresh works ; in fact only about a dozen have been added since the
Brighton Meeting, and but four or five since the MS. was placed in the printer's
hands.

The tendency of the Report has been from the first to become more and
more bibliographical. Originally it was intended to introduce nothing of a
bibliographical nature ; but experience showed that this was impossible, and
attention to such matters has been continually forced upon us. A report on
tables differs from a report on any other scientific subject in this — that
whereas in a progressive science the earlier works become superseded by

ON MATHEMATICAL TABLES. ] 65

tteir successors, and are only of historical interest, a table forms a piece of
work done, and, if done correctly, is done for all time. Thus Bsiaas, 1624,
or Vlacq, 1628, when procured, are as useful now as if the tables had been
calculated and published recently, subject to the one drawback, that it needs
a bibliographical research to determine how far their accuracy is to be relied
upon. A table is calculated for a special purpose, which purpose in process
of time ceases to be an object of practical interest, and the table is forgotten ;
but, for all that, it is the expression of a certain amount of abstract truth,
and as such is always of value, and is liable at any moment to be utilized
again for some other purpose. Thus one of the most useful objects of the
Report is to give in an accessible form accounts of old tables that have passed
out of notice, as even the most special table is never so obsolete that some
fresh use may not be found for it in the future; and it is of little value to
describe an old and unimportant work without such additional explanation as
may lead to its easy identification, with references to the works that contain
information of importance to its user.

Art. 3. But, apart from the necessity of giving bibliographical information
with regard to some works in order to render the descriptions useful, it is to
be noticed that mathematical history is practically nothing but mathematical
bibliography, as the number of letters and other manuscript documents bear-
ing upon the subject is very small. This being so, it seemed a pity when the
examination of any work showed it to possess some interest, even though of
a purely historical kind, to ignore it entirely merely because the table it
contained was clearly destitute of practical value*. The whole additional
space thus devoted to bibliography does not altogether amount to more than
a very few pages ; and the chief concession that has been made to it is in the
list of titles in § 5, where in several cases the full titlepage has been tran-
scribed. This, with one or two 'exceptions, has only been done in the case
of the tables of logarithms immediately following their invention in 1614.
An examination of a great number of works of reference in regard to this
matter has shown us how inaccurate, not only in details but even in pro-
minent facts, are the accounts usually given. With the exception of
Delambre, Lalande Xva. his ' Bibliographie Astronomique '), and De Morgau,
it is not too much to say that not a single writer on the subject is to be
trusted. Those only who have had occasion to investigate any historical
point, Uke that of the invention of logarithms, can appreciate the slight value
that was set on accuracy previously to the dawning of a more careful age at
the beginning of the present century. It is necessary to give this caution, as
any one who took the trouble to compare certain statements made in this
Eeport with those given in such works as Thomson's ' History of the Eoyal
Society,' or even Hallam's ' Literature of Europe' (founded on earlier works),
might imagine that our account involved matters of opinion and was liable
to be disputed ; whereas we cannot find that any previous Writer ever did
(or perhaps could in the then state of libraries) examine or even see all the
works relating to this period. It is also worthy of remark that the early
logarithmic tables form a most remarkable bibliographical tangle. For some
years it was customary to always place the name of Napier on the titlepages

* " It would be something towards a complete collection of mathematical bibliography,
if those who have occasion to examine old works, and take a pleasure in doing it,
would add each his quotum, in the shape of description of such works as he has actually
seen, without any attempt to appear more learned than his opportunities hare made
him."— De Morgan, 'Arithmetical Books,' p. x. See also 'Companion to the Almanac,'
1851, p. 5.

166 REPORT — 1873.

of works on logarithms, as being their inventor, and, if the logarithms were
deeimal, that of Briggs (and perhaps also that of Ylacq) in addition. Thus
the ' Arithmetica ' of 1628 will be found in bibliographies and library cata-
logues usually under the name of Napier or Briggs, and very rarely under
that of its author Vlacq. If to tiiis confusion be added the additional com-
plication produced by the varieties of ways in which the names of the three
leading logarithmic calculators were spelt, it may easily be inferred how
incorrect and confused is all the information to be obtained from bibliogra-
phical sources, whether general or mathematical*. It is on this account
.that we have thought it desirable to give the titles of these works in full in
§ 5. Perhaps it would not have been possible to see so many of them
in any one other country except this ; and the value of a number of such
titles collectively in the same list is much greater than the sum of their
separate values when scattered in difiFerent works.

Art. 4. While on the subject of bibliography, it is proper to remark that,
in the cases where the fuU titles have been given in § 5, there is a certain
slight want of uniformity in the way in which they have been transcribed,
viz. in the use of capitals, the writing at fuU length of words abbreviated,
and the modernizing the language by the substitution of u for v or i for j,
and vice versA. Titlepages are printed partly in capital and partly in Koman
and italic characters ; and when they are transcribed wholly in Koman letters,
there arise several uncertainties. Thus it is usual in the portion printed in
capitals to replace IT by V and J by I, and very often not to use a larger
letter after a full stop or for a proper name ; and in copying the whole in
Roman letters it is doubtful whether to write these as they are, or to recon-
vert them. We are inclined to think that the best plan (except when capitals
are reprinted as capitals &e., in which case no difftculty occurs) is to make an
exact copy, and not even introduce a capitel letter after a full stop, although
the author would no doubt have done so himself had he printed his title-
page in Roman characters throughout. Exception must, however, be made
in the case of proper names. These rules have not been followed out com-
pletely in one or two of the earliest titles that we copied, before experience
had taught iis that in bibliographical matters the greatest attainable accu-
racy should be invariably striven after; also one or two. abbreviations have
been replaced by the words at length' (such as e. g. " serenis'"' " by " sere-
nissimi " or " atq ;" by " atque"). Whenever, of course, any difference from
ordinary spelling is observed, it may be taken for granted that the title is so
printed in the book ; the utmost change that has been made being that some
words in a few of the titles are modernized.

The foregoing remarks apply to the titles that are transcribed at lengtk ;
but a few words must also be said with regard to those in which only
enough is given to identify the books described without possibility of mis-
take. Wherever words are left out from the title, the omission is marked

* Even Babbage makes a bibliographical error on the first page of the preface to his
tables, where he says that " the first 20,000 were read with those in the Trigonomotria
Artificialie of Briggs." The ' Trigonometria Artificialis ' was calculated by Vlaeq, and
published by him two years after Briggs's death, though the 20,000 logarithms ap-
pended were of course originally computed by Briggs. Any one who wiU look at the
title of the ' Trigonometria Artificialis ' in ^ 5 will see how easily a mistake of this kind can
be made ; and in fact an inspection of the titles of the other works of this period will show
that it would be difiicult for any one who had not bestowed some attention on the history
of logarithms to assign them to their true authors. Part of the confusion that exists is
due to Vlacq's excessive modesty, which led him on the titlepages of his works to give
quite a subordinate position to his own name compared with those of Napier and Briggs.

ON MATHEMATICAL TABLES. 167

by dots, except between place and date, where 'the publishei-'s name almost
invariably occurs ; so that, this being understood, the separation by a comma
was considered sufficient. If the work of the Report had to be performed over
again, we should adopt a set of fixed rules with regard to the use of initial
capitals in the printing of words in titles, instead of. leaving the matter to
caprice or the printer ; as it is, the treatment in this respect has been fairly
unifonn, but might have been better. Such details may seem insignificant; but
it is desirable that nothing should be regarded as arbitrary. With regard to
the number of pages assigned to books in § 5, there is also a certain want of
uniformity : at first we merely looked at the number on the last page, and
(having assured ourselves that the pagination was continuous) regarded that
as^ the number of pages, ignoring the few pages at the beginning (usually
with a roman pagination) that are devoted to preface &c. ; but afterwards
we included these also. Our object merely was to give an idea of the size
of the work ; so that (except in the cases where the interest of the book was
bibliographical, when we took pains to be quite accurate) it was not thought
necessary always to count pages that were not numbered. Sometimes it
seemed desirable to give the number of pages occupied by the tables instead
of the number in the whole book ; and in a few cases, where the pages were
not numbered, it was not considered worth while to count them, or even give
an estimate. It may be remarked that very frequently (we think we might
say more often than not) the pages on which extensive tabular matter is
printed are not numbered.

Art. 5. The distinction mentioned in § 2, art. 8, between works that are
and works that are not described in the Report, viz. that the names of the
authors of the former, when the works are referred to, are printed in small
capitals, and of the latter in roman characters, has been adhered to as carefully
as possible ; but it has been found to be very troublesome and unsatisfactory.
We have generally thought it sufficient to print the name in small capitals
only once in a paragraph ; and when there is no risk of mistake (as in the
description of the work in question itself) the name has been printed in
ordinary roman type : the distinction will not be retained in future Reports.
Also, with reference to the meanings to be attached to the words 8vo, 4to,
&c., explained in § 2, art. 9, experience has shown that it is more conve-
nient to use these terms in their technical significations, viz. as defined by
the number of pages to the sheet ; and in future Reports they will be so
used. It should be stated that, except in the ease of a few books of no
bibliographical interest, these have been the meanings actually adopted.
Care was taken that this should be so in regard to aU works of bibliogra-
phical interest ; and in most other cases the size, as estimated by the eye,
agrees with the technical signification.

Art. 6. In § 1 it is stated that the Committee had determined to print and
stereotype certain tables of e'' and e'", and of h3rperbolic sines and cosines
which had been commenced by the reporter, and that they were then in th«
press. Only four pages were set up when the above statement was written ;
and shortly afterwards, when the elliptic functions (referred to further on
in art. 16) were in process of calculation, it became clear that they would
occupy so much attention that it was not likely that the tables of e" &o.
could be continued by the reporter till after their completion, and, further,
that the publication of the elliptic functions would tax the resources of the
Committee to such an extent that it was not probable that they would have
the means of printing any thing else, at all events for some time. These
tables were therefore withdrawn ; and the reporter contemplates completing

168 KKPORT— 1873.

them (very little more remains to be done) after the publication of the
elliptic functions, when they -will probably be communicated to one of
the learned societies. The table of powers by the reporter, mentioned in
§ 3, art. 5, is entirely completed, except for the final verification by differ-
ences, which is in progress ; and the printing will be commenced very shortly ;
but as it is intended to prefix to it a list of constants, with historical notices
of the calculation of each, the publication may be somewhat delayed.

Art. 7. Any one who studies the Report attentively cannot fail to notice
differences of modes of description in it. These are only verbal, and will be
seen to be unavoidable when it is considered that, as a rule, the account of
each book was written by itself on a separate piece of paper, and that not
tiU aU had been arranged, and the Eeport was in print, was it easy to com-
pare the descriptions of the same table occurring in different works, and
therefore written under different circumstances. Very few of these " dis-
crepancies " have been removed, partly because, as each description was cor-
rect, it seemed scarcely worth whUe to make alterations for the sake of a
fictitious uniformity, and partly because we made it a rule that, a descrip-
tion having been written in the presence of the book, it ought not to be
altered when the book was absent. Slight differences of style and manner
are inevitable in a work the performance of which has extended over the
space of two years, as experience must always continually modify to some
extent both opinions and modes of thought and expression ; of course, if the
work could be done over again with the experience already obtained, the
descriptions would be more uniform.

Art. 8. An objection might be made on the ground that descriptions are given
of some very minor works, which have not even the bibliographical interest
due to age. In answer to this it is to be noted (1) that it is sometimes as
important to know that a book does not contain any thing of value as to know
what is in it if it does, and that the reader alone should be left to decide
what is and what is not valuable ; and (2) that no book is so insignificant
that in the future a correct account of its contents will not be of value.
" The most worthless book of a bygone day is a record worthy of preserva-
tion. Like a telescopic star, its obscurity may render it unavailable for
most purposes ; but it serves, in hands which know how to use it, to deter-
mine the places of more important bodies " (De Morgan, ' Arithmetical
Books,' page ii). Although the primary object of the Eeport is utility in the
present, stiU it is not desirable to entirely forget the wants of the future.
The difficulty the historian of science meets with consists not so much in
getting a sight of the books the existence of which he knows, as in finding
out the names of the second- and third-rate authors of the period he is con-
cerned with. Bibliographies grow more valuable as they increase in age ;
and it may be predicted with confidence, that long after every vestige of
claim to represent the " state of science " has passed away from this Report,
the list of names in § 5 will be consulted as a useful record of nineteenth-
century authors of tables. It might be thought that a less detailed descrip-
tion of unimportant books would suffice ; but it is only necessary to point
out in reply, that work, unless done thoroughly, had better be left alone.
An account of all the tables in a book is absolute, whereas an account only
of those that seem to the writer worth notice is relative. Want of thorough-
ness is the thing most to be dreaded in aU work of a bibliographical, his-
torical, or descriptive nature. It is this want that renders all but valueless
the greater part of seventeenth and eighteenth-century writings of this
class ; and any 9ue who performs such work in an incomplete or slovenly

ON MATHEMATICAL TABLES. 169

manner, merely accumulates obstructions which obscure the truth, and ren-
ders more difficult the task of his successors, who will have to be at the
pains not only of doing the work again de novo, but also of correcting the
errors into which others have fallen through his imperfect accounts.

Art. 9. With regard to the future Keport on the subject of general tables
that has been mentioned more than once, and is intended to be supplemen-
tary to the foregoing, it may be stated that a number of additional tables
have already been described and will be included in it ; but the cooperation
of others in the matter is requested. Whether the descriptions in the Sup-
plement win resemble those in this lleport will of course depend on the ex-
tent of the former, as, if the number of works described be large, it may be
necessary to practise some curtailment.

It is requested also that notices of errors detected in the Eeport may be
sent to the reporter (see p. 12).

Art. 10. Although, as already stated, this Eeport has no pretensions to
completeness, stiU any one who notices the non-appearance of names well
known in calculation (such as that of Legendre) is asked to read the con-
clusion of § 1, the list of articles in § 3, and enough of the introductory
matter in § 2 to comprehend clearly the spirit that has directed the selection
of works iucluded, before coming to the conclusion that the omission was not
intentional. Books such as Legendre's ' Eonctions EUiptiques ' and Jacobi's
' Canon Arithmeticus,' though forming separate publications, yet belong more
properly to a later portion of the Committee's work, as they are conclusive,
not subsidiary tables ; the former belongs to Division II., and the latter to
Division III. (see § 1, p. 4).

It is perhaps worth noting explicitly, that the word Report has sometimes
been used to donote the whole Eeport that .is contemplated by the Committee,
including the accounts of the Integral and Theory-of-Number tables, and
sometimes only the portion of it that wiU form one year's instalment ; but
the context always shows, without risk of confusion, the meaning to be
assigned.

Art. 11. It was originally intended that the list in § 5 should merely con-
tain the titles of the books described in §§ 3 and 4, with references to the
section and article where each description was given. But it has been found
convenient to render it in addition more of an index to the whole Eeport by
adding cross references, and also a few titles of papers often referred to, as
well as references to the places where certain other works or tracts (besides
books of tables) were noticed. One or two remarks that should have appeared
in the accounts of the works themselves in §§ 3 and 4 have been added
after their titles in .§ 5 (see Babbage, Noeie, 1844, and Napzee, 1619, in

§5)-

A table of contents is given at the conclusion of this postscript. Whether
a work of reference ever gets into use or not depends more on the complete-
ness with which it is indexed than on any thing else.

Art. 12. The following statistics will not be found without interest. The
number of separate books of tables described at length in this Eeport (ex-
clusive of different editions and of works only noticed incidentally) is 235, of
which only 5 are derived from second-hand sources. The 230' that have
thus come under the eye of the reporter are thus distributed among the dif-
ferent countries :^-

Great Britain and Ireland .... 109 France 27

Germany (including Austria &c.) 66 Holland 8

170 REPOR'T— 1873.

Denmark - 7 Portugal 1

Italy 3 Sweden 1

United States 3 Eussia 1

Switzerland 2 Egypt 1

Spain 1

Belgium supplying none. These figures afford no comparison between Great
Britain and other countries ; hut they give a fair idea of the relative table-
publication of foreign countries, or, at all events, of the relative proportions in
which their tabular works are to be found in English libraries. The numbers
of .tables published in some of the chief towns are as follows : — London 94,
Paris 23, Berlin 18, Leipzig 17, Edinburgh 11, Vienna 5, Copenhagen 4,
New York 3. Of the 109 works published in Great Britain and Ireland the
following is the distribution : — England 96 (London 94, Boston 1, Ci-
rencester 1), Scotland 12 (Edinburgh 11, Glasgow 1), Ireland 1 (Dublin),
showing the paramount position of London in the publishing trade in this
country.

Art. 13. GoiTTENTS OP THE Repokt that was intended to be pkesented to
THE Bradpokd Meeting, 1873. — Owing to the great amount of space already
occupied in the present volume by the foregoing Eeport, it seemed desirable
to postpone for a year the Report which it was till recently intended should
be presented to the Bradford Meeting, and only to give here a brief
description of the work performed in 1872-1873. This latter Eeport (which
is not lengthy) consists of three parts — (1) Tables of the Legendrian Func-
tions; (2) List of errors in VtAca's 'Arithmetica Logarithmica,' 1628 or
1631 ; (3) Account of the tabulation of the Elliptic Functions.

Art. 14. The Tables of the Legendrian Functions (Laplace's Coefficients). — ■
These give P"(ar) to n=7 from a;=0 to x=l at intervals of -01, viz. the
functions are : —

F' = l,

F' = i(3a^-l),

P' = |(5a;'-3a;),

P' = i(36a;*-30ar'4-3),

r'=^(63ar'-70x' + 15x),

P' =TV(231a?'-31.5a;''+105d;»-5),

P' = TV(429a?' - 693a;' + 315a;' - 35a;) ;

and as only powers of 2 appear in the denominators, all the decimals ter-
minate, and their accurate values are therefore given. The work was per-
formed in duplicate-*- one calculation having been made by Mr. "W. Barrett
Davis, and the other under the direction of the reporter, by whom the two
were compared, the errors corrected, and the whole diflferenced. As the
accurate values of the functions were tabulated, the verification by diflFer-
ences was absolute. A short introduction on the use of the tables in inter-
polation was written by Prof. Cayley, who has also made drawings of the
curves y=F"(a;) over the portion calculated.

Art. 15. The List of Errors in Vluec^s 'Arithmetica Logarithmica' (1628
or 1631). — It seemed very desirable that a complete list of the errata in
ViAca, 1628 or 1631, should be formed for the convenience of those who
have occasion to employ ten-figure logarithms. No less than five copies of
this work have been continually in use in the calculation of the Elliptic

ON MAtflEMATICAL *ABI.E8, 171

Functions (see next article) during the last year ; and it is the ten-figure
table chiefly used. Besides this, the errata in ViAca are known with more
certainty than are those in Vega, 1794.

This list had only been partially formed when it was determined to post-
pone the Report ; and it is believed that the year's delay may possibly result
in its being made more complete- It is proposed to add a list of errata also
in Dodson's ' Antilogarithmio Canon,' 1742 (§ 3, art. 14), and perhaps to
consider the subject of errors in tables generally.

Art. 16. The account of the Tabulation of the EUvptic Functions. — In Sep-
tember 1872 it was resolved to undertake the systematic tabulation of the
Elliptic Functions (inverse to the Elliptic Integrals), or, more strictly, of
■ the Jacobian Theta Functions which form their numerators and denomi-
nators.

The formulae are : —

= 1 — 2ocos2a;+22*co8 4a;— 25°co8 6iK+. . .,

^ 2K.i; 1 ^2Kx

1/1 9 25 S

= JL(23'*sina;— 22Tsin3a'-l-2g'<' siuSa?— . . . ),

= ( - ) (%* cos x+2^ cos 3.r +22 ^cos 5*+ . . .),

= fc'* (1 H- 22 cos 2ar + 22* cos 4a7 -\- 2(( cos 6a; + . . .) ;

so that

•i^ix „ 2K^ . _2Ka;
sin am = Wi -r w ,

TT T IT

2K:a; ^ 2Ka; . „2Ka;
cos am =03 -T-o ,

T IT T

2K«; „ 2Ka; . „2K,r
A am ="»• "^t* >

IS It T

q being, as always, e « ; and the tables, when completed, will give
e, e^, e^, e, and their logarithms to eight decimals for

i» = 1°, 2^ . . . 90°, A: = sin 1°, sin 2°, . . . sin 90°.

The tables are thus of double entry, and contain eight tabular results for
each of 8100 arguments, viz. 64,800 tabular results The arrangement wiU
be so that over each page h .shaU be constant ; and at the top of each page
certain constants (i. e. quantities independent of x\ such as

K,K',J,J',E,fc*,(p)%(i)',?,&c-,

172 REPORT— 1873.

and their logarithms, which are likely to be wanted in connexion with the
tables, win be added. K and K' (complete eUiptio integrals) were, as is weU
known, tabulated by Legendre, and published by him in 1826.

For the performance of the calculation of 9 and 63 (9, being deduced from
0) 8500 forms were printed and bound up into 15 books (550 in each, with a
few over). Each book, therefore, contains forms for the calculation of six
nineties, viz. from Jc=Bm a° (say), .r = 0°, to Z; = sin (a°+ 5°), a; = 90°. Similar
forms for the calculation of 9^ and 0^ were printed and bound up into 15
other books.

The work has been in active progress since the beginning of October 1872 ;
and eight computers have been engaged from that time to the present, under
the superintendence of Mr. James Glaisher, F.E.S., and the Eeporter. About
three quarters of the work is now performed — 9 having been calculated com-
pletely, and its accuracy verified by differences, and 9, being nearly finished
also, while very considerable progress has been made with 9, and 9^.

It is intended tha,t the tables, which will be completed, it is hoped, by
February 1874, shall form a separate work, and that they shaU bo preceded
by an introduction, in which all the members of the Committee will take part,
— an account of the application of the functions in mathematics generally
being undertaken by Professor Cayley, of their application in the theory of
numbers by Professor H. J. S. Smith, and of their use in physics by Sir W.
Thomson and Professor Stokes, whUe the account of the method of calcula-
tion &c. win be written by the Reporter.

The magnitude of the numerical work performed has not often been ex-
ceeded since the original calculation of logarithms by Briggs and Vlaeq,
1617-1628 ; and it is believed that the value of the tables wUl be great.

After the circular and logarithmic functions there are no transcendants
more widely used in analysis than the Elliptic Functions ; and the tables wiU
not only render the subjects in which they occur more complete, but wiU also,
to a great extent, render available for practical purposes a vast and fertUe
region of analysis. Apart from their interest and utility in a mathematical
point of view, one of the most valuable uses of numerical tables is that they
connect mathematics and physics, and enable the extension of the former to
bear finiit practically in aiding the advance of the latter.

Art. 17. Note on the Centesimal Division of the Degree. — In the note
on p. 64 we have expressed an opinion that Briggs and his followers, by
dividing centesim ally the old nonagesimal degree, showed a truer appreciation
of how far improvement was practicable, or indeed desirable, than did the
French mathematicians who divided the quadrant centesimaUy. On reading
Stevinus's ' La Disme,' the celebrated tract in which the invention of decimal
fractions was first announced, we found that the centesimal division of the
degree was there suggested. The following extract from ' La Disme ' is
taken from pp. 156 and 157 of ' La Pratique d'Arithmetique de Simon Stevin
de Bruges ' (Leyden, 1685), near the end of which ' La Disme ' appears in
French. The first publication of the tract, as far as we can find, was in
Dutch, under the title " De Thiende .... Besohreven door Simon Stevin van
Brugghe " (Leyden, 1585).

" Article V. Des Computations Astronomiques. — Aians les anciens Astro-
nomes parti le circle en 360 degrez, ils voioient que les computations Astro-
nomiques d'iceUes, auec leurs partitions, estoient trop labourieuses, pourtant
ils out parti chasque degre en certaines parties, & les mesmes autrefois en
autant, &c.,^ fin de pouuoir par ainsi tousiours operer par nombres entiers, en
choissisans la soixantiesme progression, parce que 60 est nombre mesurable

ON MATHEMATICAL TAH'LES. 173

par plusienrs {sic) mesures entieres, h sgauoir 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,
mais si Ton pent eroire I'experience (ce que nous di'sons par touts reuerence
de la venerable antiquite & esmeu aueo I'vlilite commune) certes la soixan-
tiesme progression n'estoit pas la plus commode, an moins entre celles qui
consistoient potentiellement en la nature, ains la dixiesme qui est telle : Nous
nommons les 360 degrez aussi Commeneemens les denotans ainsi 360(0) *
& chascun degre on 1(0) se diuisera en 10 parties egales, desquelles chascune
fera 1(1), puis chasque 1(1) en 10(2), & ainsi des autres, oomme le semblable
est faiot par plusieurs fois ci deuant "f-

At the end of the ' Appendice du Traiot© des Triangles,' which concludes
the fourth book of the " Cosmographie " in Albert Girard's edition of
Stevinus's collected works, Leyden, 1634 (p. 95), there occurs the following
note : —

" Notez. — -Pay descrit un chapitre contenant la maniere de la fabriqiie &
usage de la dixiesme progression aux parties des arcs avec leurs sinus, & de-
clare combien grande facilite en suit, comparee h la vulgaire soixantiesme
progression, de 1 deg, en 60(1), & 1(1) en 60(2), &e. laquelle matiere pour-
roit ici sembler requerir sa place : Mais veu que les principaux exemples
d'iceUe se prennent des cours moyens des Planetes & autres comptes communs
avec iceux, qui jusques ici ne sont point encores descrits, nous avons applique
le susdit chapitre derriere le traicte d'iceUes Planetes, h sgavoir en l'Aj>;pen-
dice du cours des Planetes."

To which is appended the following note by Girard : — " Ceste promesse ne
se trouve pas avoir este eifectuee."

Steichen, in his ' Memoire sur la vie et les travaux de Simon Stevin '
(Brussels, 1846), p. 62, says that Stevinus promises a chapter on the manner
of constructing a table of trigonometrical lines " pour la division de la cir-
conference en parties decimales." This is not correct, as the quotation
from ' La Disme ' shows that Stevinus's idea was to divide the degree cen-
tesimally.

Briggs, in the * Trigonometria Britannica' (p. 1), states that he was led to
divide the degree centesimaUy by the authority of Vieta (" Ego vero adductus
authoritate Vietse, pag. 29. Calendarij Gregoriani, & aliorum hortatu,
Gradus partior decupla ratione in partes primarias 100, & harum qu.amlibet
in partes 10. quarum quselibet secatur eMem ratione. Atque hse partes cal-
eulum reddunt multo facilorem (sic), & non minus certum "). We have
looked through ' Erancisci Vietse Eontenasensis .... Eelatio Kalendarii vere
Gregoriani. .. .1600" (Colophon: ' Excudebat Parisiis . . . . ,' 40 leaves, as
only the rectos are numbered, 1 to 40) without finding, either on p. 29 or
elsewhere, any mention of the division of the degree. Without venturing to
say that there is nothing of the kind in the book, it is not unlikely that the
wrong work of Vieta's is referred to, as we have found many other seven-
teenth-century references inaccurate; and this is rendered more probable
when it is remembered that the ' Trigonometria Britannica ' was published
after Briggs's death.

But granting, as is likely, that Briggs did derive the idea from Yieta, it is
very probable that the latter himself obtained it from Stevinus, and perhaps
adopted it without acknowledgment, as unfortunately it is to be feared that

* Stevinus encloses tne exponential numbers in complete circles, for which we have
throughout substituted parentheses, for convenience of printing.

t This refers to the preceding articles of the ' Disme,' where the decimal division is
explained.

174 REPORT— 1873.

Vieta was bigoted enough to suppress the name of a heterodox author,
such as in all likelihood' Stevinus was. There can therefore be but little
doubt that the original suggestion for the centesimal division of the degree is
contained in the sentence quoted from ' La Disme ; ' but we intend to inves-
tigate the question further, and endeavour to decide it conclusively.

CONTEITTS OF PaBX I. (1872 AND 1873) OF THE KbPORT ON MATHElIATIC.Ui-

Tables.

Page

§ 1. General Statement of the Olrfects of the Committee 1

§ 2. General Introdiiction to the present Beport, and explanation of its Arrangeinent
and Use.

Art. 1. (Beport includes general tables ; see also conclusion of § 1) 4

2. (Object of the Report) 5

3. (Previous works on the subject of tables ; bibliographies, &o.) .5

4. (Mode of arrangement of the Report ; meaning of a prefixed asterisk) 7

5. (Explanation of the marks, conventions, terminology, &c. adopted) 8

6. (The particular edition of a work described is arbitrary) 10

7. (The tables themselves, and not merely their titlepages, have been ex-

amined) 10

8. (Why certain names are printed in small capitals, or enclosed in square

brackets; see also § 6, art. 5) 10

9. (Use of the words 8vo, 4to, &c. ; see also § 6, art. 5) 11

10. (Libraries consulted) 11

11. (The Beport is imperfect ; information is asked from persona possessing

knowledge on the subject of tables) 12

12. (Traverse tables omitted) 12

13. (Errors in tables) 13

14. (The works are described from inspection ; care taken in preparation of the

Beport) 14

§ 3. Separate Tables, arranged according to the nature of their contents ; with
Introdiictory Remarks on each of the several kinds of Tables included in the
present Beport.

Art. 1. Multiplication tables 15

2. Tables of proportional parts 20

3. Tables of quarter squares 21

4. Tables of squares, cubes, square roots, and cube roots 25

5. Tables of powers higher than cubes 29

6. Tables for the expression of vulgar fractions as decimals 30

7. Tables of reciprocals 33

8. Tables of divisors (factor tables), and tablesofprin.es 34

9. Sexagesimal and sexcentenary tables 40

10. Tables of natural trigonometrical functions 41

11. Lengths (or longitudes) of circular arcs 47

12. Tables for the expression of hours, minutes, &c. as decimals of a day, and

for the conversion of time into space, and vice versd 48

13. Tables of (Briggian) logarithms of numbers 49

14. Tables of antilogarithms 62

15. Tables of (Briggian) logarithmic trigonometrical functions •.. 63

16. Tables of hyperbolic logarithms (viz. logarithms to base 2-71828 ... ,) 68

17. Napierian logarithms (not to base 2'71828 ... .) 70

18. Logistic and proportional logarithms 73

19. Tables of Gaussian logarithms .- 75

20. Tables to convert Briggian into hyperbolic logarithms, and vice versa 78

21. Interpolation tables , , 79

1 22. Mensuration tables 79

23. Dual logarithms 80

ON COAL-CUTTING MACHINERY. 175

24. Mathematical constants , , 81

25. Miscellaneous tables, figurate numbers, &o 83

j4. WorJcs containing Collections of Tables, arranged in alphabetical order 85

i 5. List of Works containing Tables that are described in this Beport, with references

to the section and article in which the description of their contents is to be found 143
) 6. Postscript.

Art. 1. (Report is that presented to the Brighton Meeting enlarged) 164

2. (Alterations since the Brighton Meeting; Report has been made more

bibliographical) 164

3. (Reasons for introducing bibliography ; inaccuracy of previous writers) ... 165

4. (Explanations with regard to the list of books in § 5) 160

5. (Supplementary explanations referring to § 2, arts. 8 and 9) 167

6. (The tables of hyperbolic antilogarithms and powers calculated by the Re-

porter; 1 1, and § 3, art. 5) 167

7. (Slight differences in mode of description observable in the Report) 168

8. (Why some unimportant works are included) 168

9. (The Supplementary Report on general tables) 169

10. (Some books omitted intentionally, as belonging more properly to subse-

quent Reports) 169

11. (§5 has been made au index as well as a list of titles of books) 169

12. (statistics with regard to books described in the Report from mspection) ... 169

13. (Contents of the Report that was intended to be presented to the Bradford

Meeting) 170

14. (The tables of the Legendrian functions) 170

15. (The list of errors in Vlaoq, 1628 or 1631) 170

16. (The account of the tabulation of the elliptic functions) 171

17. (Note on the centesimal division of the degree).. 172

ERRATA.
Page 6, line 8. from bottom, for VoggeaioS read Poggendorff.
Page 16, line 25 from top, for multiplication read multiplication table. .

Observations on the Application of Machinery to the Cutting of
Coal in Mines. By William Pirth, of Birley Wood, Leeds.

[A Communication ordered by the General Committee to be printed in extenso.]

The object of this paper is to submit for consideration some matters touob.-
ing the history of the now more than ever absorbing subject of cutting coal
in mines by mechanical means.

It is intended to avoid all technical and scientific symbols, and to convey,
in the most simple manner, whatever information is at my command, and to
give, from practical experience, spread over long periods, the results derived
therefrom, and to show that machinery can be, and is now, applied to the
purpose equally to the advantage of the masters and of the men.

I am aware that there are now several distinct modes of doing the work,
and doing it well ; but it is not in my power to give any reliable information
upon the competitive status which the successful machines hold towards
each other. I shall therefore have in this paper to confine myself more
particularly to the introductidn of coal-cutting machinery driven by com-
pressed air, and the results obtained from the invention now known as
"Firth's Machine," which was unquestionably the first that ever succeeded
in reducing to actual practice the cutting of coal in mines.

176 EEPORT— 1873.

When the severe nature of the employment of manual labour for the
" hewing of coal " and the great dangers which beset that occupation are
taken into thoughtful consideration, it is not surprising that much sympathy
should have been always excited in favour of the coal-worldng class. All
men who have thought upon the subject have felt a strong desire that some
mechanical invention might be made to ameliorate the severe conditions
of that occupation.

The statistics of the comparative longevity of the working classes show
that the duration of the lives of colliers (apart from special accidents) is
lamentably low ; and as respects the " hewers " or " pickmen," whose work is
the most exhausting, they must especially, and in a large degree, contribute
to, and account for, much of that average shortness of Ufe.

The really hard work of a colliery falls upon the " hewers ; " and the effect
is very often to stamp the men with the mark of their trade, and (through
the constrained position of their daily toil) to alter and distort many of the
more delicately formed persons ; and it is due to these men as a class, that
their weaknesses should be mildly judged, having regard to the scanty oppor-
tunities hitherto afforded to them for intellectual culture, and the unequal
sacrifices which press so heavily upon them in the most valuable and im-
portant branch of all our industries.

In 1862 some experiments were commenced at West Ardsley, by the em-
ployment of compressed air, to devise a cutting-instrument in the form of a
pick. It was to be moved on the face of the coal, striking in a line and with
such force as would out a groove deep enough to admit of its being easUy
taken out. In the early stages there were many and serious discouraging
symptoms discovered, but on the whole we were well satisfied that they could
be overcome by perseverance. We set about to improve the defects, and
battle with the difficulties as they presented themselves; and after some
years we were in possession of a coal-getting machine, in combination with
air-power, more suitable for the performance of the work which we had
undertaken than we ever anticipated.

Much surprise has been expressed at our slow progress during the ten
years which have elapsed since the time when we believed that we had
reached success ; but when the peculiar circumstances which surround the
work, and the nature of the work to be done, are taken fairly into account,
the delay need not excite any astonishment. It weis in many respects a
new field to be broken up, and accompanied by numerous uncertainties.
It has been more or less so with most of the important inventions which
have gone before it; indeed the steam-engine, whose origin cannot be
traced back, was known as a prime mover nearly two centunes before it was
sufficiently developed to be recognized as a valuable machine.

We found, however, that we had to contend against much prejudice and
resistance. Those who were the most likely to be benefited by it were
either openly hostile or manifested an unfriendly disposition towards the
machine ; and, added to these embarrassments, we failed to obtain any
general encouragement from those who exerted the greatest influence over
the coal-mining interests of the country ; but through the recent dearness
of coal, the attention of the country has been drawn to the subject, the
pubUc mind has been powerfully impressed with the necessity for some
improved means of working the mines, and coal-cutting machinery is now
universally looked to as the principal source from which relief is to come.

From the altered feelings of the miners as to the number of hours which
they consider to be sufficient for their labour, and with the new restrictions